What type of armor should I use as a thief?

It is dependant on your build.

I am playing a Khajit thief and I use light armour. Minor spoiler below:

The thief guild provides you with some nice light armour.

The benefits of light armour are:

• You are less detectable in lower levels with light armour (the perk that light armour weights nothing is at skill level 50, in heavy armour, it is 70). As a thief you don't tend to get hit often, so leveling your armour takes longer than you would expect.
• Windwalker is great for a thief, your stamina regenerates 50% faster if you are wearing all light armour. You need to be agile so stamina is pretty important for you running away, positioning etc.
• Light armour still gives a decent armour rating, so if needed you can go toe to toe without taking a huge beating.
• Fits roleplaying.

The benefits of heavy armour are:

• Higher armour rating
• Tower of strength stops you getting knocked back if you get caught off guard. (You are a bad thief if you get caught off guard ;) )
• You can still sneak effectively once you hit level 70 and get the conditioning perk.

Using robes is also a very viable option, depending on your character set up. If you decide to go this route, I'd recommend going with illusion + alteration.

The benefits are:

• Decent armour rating with alteration stone flesh skins, combined with mage armour. (Triple the effects)
• Robes give +x% magica regen and -x% spell cost on x tree.

This means you can benefit more from the magic attributes below:

• Decent magic resistance from the alteration skill tree (up to 30% magic resistance)
• Alteration grants you the detect life spell. This is awesome as a thief, totally reduces the chances you have of getting caught off guard.
• Illusion skill tree (at level 50) lets you cast spells silently. This is perfect for a stealth character.
• Illusion grants you the invisibility spell, perfect for a thief.
• Illusion gives you crowd control for when the brown stuff hits the fan. You can make a single target (or groups at higher levels) stop fighting, fight each other, or run away.

Let’s just end the Skyrim assassin build armour debate once and for all, I won’t give monetary values as it is recommended that you keep all the armour pieces because getting extra’s if you lose any is EXTREMELY difficult especially with the advanced sets (Nightingale; Guild Master’s and Ancient Shrouded).

Okay, so first we’ll look at Shrouded Armour vs. Thieves Guild Armour (the easiest to acquire):

                  Armour  Weight   Bonus

Shrouded Cowl         13       2   +20% bow damage
         Armour       29       7   +50% poison resistance
         Gloves        8       2   +100% backstab damage
         Boots         8       2   Muffled movement

Thieves Guild Hood    15       1½  +10% better prices
              Armour  33       7   +20 carry weight
              Gloves  10       1   Lockpicking is 15% easier
              Boots   10       1½  Pickpocket success is 15% better

All in all, Guild Armour is better for thief-oriented classes who will use Lockpicking and Pickpocketing more, whereas Shrouded Armour is much better for assassin classes. Shrouded Gloves remain valuable even when you’re a high level (if you miss the Ancient variety).

Now a look at the Shrouded Armour’s alternate counterparts, the Dark Brotherhood offers Ancient Shrouded Armour (only if you kill Gaius Maro in a major city and use Olva’s Token when talking to Olva, then retrieve it from a secret room in a ruin) which has higher armour values and much higher enchantments. It also offers Shrouded Robes and Worn Shrouded Armour. The Robes are better for mage-assassins and the Worn variety is unenchanted, meaning you can get whatever enchantments you want on it. There is also the Jesters clothes and Cicero’s clothes. These are excellent if you are an assassin who uses alteration spells instead of armour or are a stealthy mage looking for stealthy gear.

                  Armour  Weight   Bonus

Shrouded Hood          0        ½  25% better sneaking
         Robes         0        ½  -15% Destruction spells cost
         Hand Wraps    0        ½  ×2 backstab damage
         Shoes         0        ½  Muffled movement  

Ancient Shrouded Cowl 19       1   +35% bow damage
               Armour 41       5   +100% poison resistance
               Gloves 15       1   ×2 backstab damage
                Boots 15        ½  Muffled movement

Jester's Hat           0        ½  +30% to sneak
         Clothes       0       1   Prices 12% better, +12% one-handed damage  
         Boots         0        ½  Wearer is muffled and moves silently
         Gloves        0        ½  ×2 one-handed damage in sneak attacks 

Cicero's Hat           0        ½  +35% to sneak
         Clothes       0       1   Prices 20% better, +20% one-handed damage
         Boots         0        ½  Wearer is muffled and moves silently
         Gloves        0        ½  ×2 one-handed damage in sneak attacks

So now we come to the most debated armour’s in Skyrim, the Nightingale Armour. I’ve met people who only use this because it looks good, I’ve also met people who hate it because they got it at a low level or don’t use the skills it gives you. People thought they’d get an upgraded Thieves Guild Armour, but Nightingale Armour is designed for Nightblades, stealthy mage-assassins who can hold their own in a swordfight. It is a mixture of combat, thief and mage armours. Well, here are it’s values at the highest level, 32 or above when receiving the quest ‘Trinity Restored’, not when you get the armour! The armour stats don’t change but the enchantments get stronger.

                  Armour  Weight   Bonus

Nightingale Hood      15       2   -17% to Illusions cost  
            Armour    34      12   +40 stamina, +50% Frost Resistance
            Gloves    10       2   +25% to Lockpicking, +25% one-handed damage
            Boots     10       2   Wearer is muffled and moves silently

Note that this is the highest level muffle achievable without crafting your own boots. The value between level 32+ Nightingale Boots and level 19+ (second highest) is over 5000 septims, meaning that at the highest level the Boots are the most valuable thing a Nightingale wears. This also makes the Nightingale Boots one of the best boots for ANY stealthy character.

And finally we reach the only Thieves Guild Armour that looks different to the standard and has better enchantments. The Guild Master’s Armour, NPC’s that wear are senior Guild members like Vex (Master Picklock), Delvin (Master Sneaker), Brynjolf (Second-in-Command) and Mercer Frey, the Master of the Thieves Guild. This armour can only be received after completing the main Thieves Guild Quest Chain and all the Quests that restore the Thieves Guilds influence in the Major Holds, you get these quests of Delvin and Vex after completing enough odd jobs in a particular Major Hold. Linwe’s Armour is encountered during one of these quests, ‘Summerset Shadows’.

                  Armour  Weight   Bonus

Guild Master's Hood   16       3   Prices are 20% better
               Armour 38      10   +50 to carrying capacity
               Boots  11       2   Pickpocket success is 35% better
               Gloves 11       2   Lockpicking is 35% easier

Linwe's Hood          16       2   +15% bow damage  
        Armour        31       6   +15 to stamina
        Gloves        11       2   +15% one-Handed damage
        Boots         11       2   +15% to sneaking

Pick whatever armour suits your character the best, or make your own. This is just to let you see the stats of all the different Thief/Assassin armours before you spend an hour getting some stuff you won't use.

I don't use full, complete sets, but use bits from different sets. I use light armor, mostly. Heavy armor, if you're not skilled with it, it will never get you anywhere as a thief or an assassin. You don't need to worry about having a high armor rating if your good at staying unseen.

Here is my set up:

Armor

A shrouded hood, can be obtained by doing the start up quest for the Dark Brotherhood (choose either side) and while you're in the sanctuary, its somewhere :P (its been a while) if you do decide to join the DB I suggest using that until you kill or save Cicero and get it then. The shrouded hood adds 25% better sneaking. Cicero's hat makes you 35% better at sneaking. Extremely useful. For my boots, I stick with any of them that make you move silently. The one I use, is DB, so another use of DB for great sneaking. For my gloves, I wear the (yet again, DB stuff saves the day) and use the DB gauntlets. Now for the fun part of my armor. for my chestplate... brace yourself... I use Ebony Mail. Yes, yes I know I made a point of how heavy armor (such as Ebony Mail) - I strictly said it was a bad idea. But this Daedric artifact is a totally different story. It makes you pitch black while sneaking (which if you go through the coding carefully, you will find). It makes you a little better at sneaking, during the day out of shadows. but at night, or in shadows yo get about 15%. in addition, unlike most heavy armor, almost silent. it has one more irresistible trait.

As an assassin, thief, or someone who just likes to sneak around, you will notice that with low armor ratings if you get caught in the middle of an overwhelming situation, which you can't control, cause you're not sneaking, and daggers (I will get into weapons) don't really do much if you're not hidden. But this armor, if you're being attacked, will poison all nearby hostiles. To top all this off, I have an insanely good ring. I found looting some ruin, a peerless ring of sneaking, which means I sneak, 40% better.

Weapons

I use Mehrunes' Razor, with about the same base stats of a Daedric dagger, its one hell of a weapon with its small chance to kill something in one hit. I use this with my dragon bone dagger (craftable with DG DLC). For skills (perks) make sure to get 5/5 of the stealth skill (do remember, this checkpoint wont be reached until you are a level 80 sneaker, which will take up many hours of your life) make sure to get assassins blade perk, and basically focus yourself on the sneak tree in general.

Powers

Nightingale embrace shadows power is spectacular. Also, I do recommend becoming a vampire (lord is better, just because a last resort for a thief or assassin (mainly assassin) might need an extra push) if you don't like joining the vamps in DG just ask Serana after or during a certain point in the questline to bite you. (Note that a vampire/vampire lord doesn't suit everyone, but comes with major upsides.) As a vampire, you gain an active effect that makes you 25% harder to notice. Also, if you haven't had blood in a while, it grants you a crazy good power that lets you be invisible (with some other stuff I cant recall at the moment) with the shadow stone, and the NG power I mentioned you can go completely invisible 3 times a day. For shouts, Throw Voice.

Misc. Items and Other Tips

I really recommend invisible potions I believe its Chaurus eggs and Ice Wraith teeth that you can make your own with (you can get an abundance of both these things early on at Jorrvaskr in Whiterun). It's where the companions live, just look around. anyway that's what I do, you can literally sneak past everything with all that. Have fun.