If I buy Garry's Mod, will I need other games to play it?

Technically - you do not need to purchase anymore games. However, you will need to purchase and download (possibly) Counter Strike: Source and Team Fortress 2 for a lot of the textures and other game objects.

From the Steam page for Garry's Mod:

Please note: Some multiplayer servers and game-modes might require you to own other games - such as Counter-Strike: Source and Team Fortress 2.

This Reddit also provides some more details.

Counter Strike: Source is currently $19.99 at posting. Team Fortress 2 has been free since June 23, 2011. It's important to note that some others games that may be needed are basically any game that run on the Source Engine developed by Valve. This includes games like:

• Left 4 Dead
• Left 4 Dead 2
• The Half Life 2 Series
• Counter Strike: Global Offensive

...and many more! But the majority of the textures are handled with just Counter Strike: Source. I would say at the minimum, you will need that game (and you might as well get Team Fortress 2 since it's free). A complete list of Source games can be found here.

By not having the correct games, you could potentially be missing the textures needed to play maps created by others. You will see a lot of checkered purple and black objects, and actual "ERROR" messages like this:

You don't have to have any other games in order to actually play GMod.
But it is highly recommended to have HL2 + ep1/2 + CSS + TF2 in order to:

• use their maps and content
• use addons or mods requiring those games

GMod is more fun with having more stuff at your disposal. A basic installation with no other resources lacks of almost everything when it comes to maps, props, npcs, etc.

In addition to the previous answers, it is worth noting that some maps or mods have models/textures compiled with them. This means that when you download (for example) a map, even though it may have (for example) HL2 Episode 2 textures/models, you are still able to see them properly without the game installed because you have downloaded them with the map. Some addon/level creators have even used custom textures and models that you will not find in any other game.

Note that most maps do not use this method, as it increases file size and (possibly) decrease optimization of the map, making it run slower.

From my browsing of the workshop, I have found that usually maps require Counter Strike: Source or Half Life 2: Episode 2.

Without the proper textures/models installed, you will (as shown by Timmy Jim) Errors and purple/black checkered patterns, or textures with other "funky" texture issues, relating to a HDR skybox.

The other games are not required to download a map/addon, and upon joining a server with a map with these things that you do not have installed, you will still continue to download the map, however errors will still arise (and you may get script errors, but these usually aren't anything to worry about).

EDIT #1: Here is the "funky" texture thing I was talking about: Also, there is a Trouble in Terrorist Town (TTT) map called ttt_forest_final. It has HL2 Ep2 models (and maybe textures and sounds) compiled with it. Even though I have Ep2 on my computer, when my friends play it they see the same things as me. A great map that I reccomend for TTT.