It's a side mode that extends the lifespan of the game, much the same way as Mercenaries does. It's different to Mercenaries in that it's not against the clock, but it's more about reaching a location in the level. As you don't have the time mechanics at play from Mercs, you are free to go about killing the enemies as you like. There's really no "right" or "wrong" way to play it, you are given the relative freedom to develop a play style and simply refine it until you are happy.
You can use any character (provided you unlock them) you like and equip whatever weapon you like. Anything you find in the game, you can use however you want. The character choices are largely cosmetic and don't impact your play style as much as it does in Mercs, where characters have very distinct advantages/disadvantages over others. The only difference character choices make in Raid are about which weapons you think you want to use and which, of those, you want to be able to reload quickly.
Most of the customization that you'll be doing will be fitting your guns with different kinds of attachments. You can unlock many attachments in Raid and they all give your gun significantly different characteristics. For example, if you like a gun but find it empties it's clip quickly, you can add an Extended Magazine to it so you can cut down on reloads. If you're using an Machine gun SMG, you might like how quickly it takes out individual enemies, but you'll notice that even when enemies are lined up, you only take out one at a time. If you equip Piercing 2, you can take out three or so enemies, provided they are lined up so you can really cut those mobs of enemies down to size. You can modify a weapon to make it a very specialized tool or you can modify it so it becomes a good all round weapon to rely on.
Well, I suppose it's time to get kidnapped again...