Upgrades give you special powers and abilities to make yourself and your homies stronger, better, and faster. This is the equivalent of "leveling up" in a role-playing game, and generally serves the same purpose.
All upgrades require a certain respect level, and most have an associated cost in cash. You'll naturally acquire both respect and cash over the course of the game; if you're lacking cash, you can just sit and wait for your hourly income to accumulate, and if you're lacking respect, run some missions or do some diversions, Activities, or story missions.
Upgrades have nothing to do with Achievements or Trophies, so don't feel pressured into buying everything. Some of the later upgrades are incredibly expensive, and it's doubtful you'll be able to afford everything by the time you're done with the story unless you do a lot of waiting around or play Activities over and over. You should buy the upgrades that interest you or are the most helpful. For example, you may reach the point where you never fire your pistol, so buying any of the Ammo - Pistol Combat Upgrades would just be a waste of your resources.
The game is setup so that buying upgrades is practically required. You might be able to get away without them on Easy difficulty, but it's almost necessary for Normal difficulty, and it's definitely necessary for Hardcore difficulty. The most important upgrades you can get are the Damage Upgrades, which reduce how hard you're hit by enemy fire. For example, the single best upgrade in the game is arguably Damage - Bullet 4, which makes you invincible to enemy bullets. If you have only that upgrade, grenades and such may still hurt you, but even STAG's laser weapons won't make a dent in your health. Always prioritize Damage Upgrades, especially the bullet upgrades, and you'll maximize your chance of succeeding in missions.
Note that there are no upgrades for protecting your cars. You can upgrade individual cars at Rim Jobs, but you can't simply buy an upgrade that makes any given car suddenly start to operate like a tank. The only upgrade that applies to any given car you jump into is Skill - Nitrous, located in the Vehicles Upgrade, and that has limited use.
There are three types of upgrades that are automatically given, and therefore not covered here: Customization Items, Strongholds, and Activities. These are unlocked by getting through certain story missions and making certain choices during the campaign. Although they will be listed in your phone once you acquire them, they are otherwise invisible and therefore not listed here in the wiki.