For me

Bare minimum:

Pump, patch kit, 1 spare tube + levers - stored in a quick release seat bag
Helmet (actually there are occasions I've forgone a helmet - but it's very rare)
Rear blinkie (light)

Almost always necessary:

Sunglasses (or ski goggles depending on the weather )
Gloves
Reflective leg band (out of habit more than necessity but I do think it's a great all-round visibility aid)

Assuming I need to leave the bike:

U-Lock + cable

Riding in low-light/dark:

Helmet lights, front light(s) depending on how dark it is.
Reflective sash or vest

Longer rides:

Water bottle or two (with water!) + snacks
Extra spare tube
Multi-tool

I always have a computer with cadence on my bikes too but it's definitely not necessary.

A bell is very handy if commuting on trails that will have a lot of traffic. I've recently been experimenting with a mirror too. I will also often use my helmet mounted camera (I have a Contour HD), especially in low-light conditions.

I don't have racks or permanent fenders on most of my bikes. For rides < 10 mi I don't think a backpack is all that uncomfortable, and I wear one for roughly 1/5th of my commutes.

Once you start riding more, it might be worth getting bike shorts. Even the cheaper $20 liner style ones you can wear under jeans are a good start for commuting, imo, and very serviceable ones can be had for <$35. For jerseys, just having a technical fabric base layer is fine for starting out - any cheap synthetic athletic shirt from Target/Old Navy will work to keep you a lot drier.

I feel like when you go to the trouble of having a helmet, rear blinkie and a reflective leg band (in any light conditions), drivers realize and understand that you are trying to be visible and safe and that you're neither a daredevil fixie-riding messenger-hipster nor a total slowpoke n00b (even if you are one of those!).