How do I increase my inventory space in Borderlands?

Collecting a Backpack SDU will increase your inventory by three slots, and are rewards for completing the claptrap rescue missions. There are 13 all together, not including the first one. It should be noted you're not guaranteed to get one for each mission on your second play through. The maximum inventory size is 72 slots.

Here's a complete list of locations with a claptrap mission (source):

• Sledge's Safe House
• Lost Cave
• New Haven
• Tetanus Warren
• Earl's Scrapyard
• Krom's Canyon
• Trash Coast
• The Lumber Yard
• Old Haven
• The Salt Flats
• Crimson Fastness
• Lockdown Palace
• Tartarus Station

Also, as @Weslei pointed out, if you buy Mad Moxxi's Underdome Riot you will also get a bank to store items in.

The easiest way to increase storage space in your inventory is to get backpack storage deck upgrades. These are typically handed out when you complete a "Repair a claptrap" mission. So, as you're wandering about, be sure to look and listen for broken claptraps!