Congestion won't be eliminated, it might even increase. Having self driving cars can in theory make far better use of existing roads (see ratchet freak's answer for example).
However currently population is going up, and desire to spend on infrastructure is not going up. Assuming those trends continue you will start seeing reduced congestion as self driving cars are adopted. That'll make people happy, but it won't last. Population increase will start eating into the gains, and unless we start spending more on roads congestion will start to raise again.
I would even guess that as long as the amount of congestion climbs slowly it will be allowed to climb above current levels because it is much more productive time:
- eat a meal
- brush your teeth
- watch TV (or for long commutes a movie)
- if you own an RV (small RVs might get more popular):
- take a shower
- get dressed
- knowledge workers could do some of their job during the commute (what do you think Apple/Google/other employees do on the buses now?)
I'm also not 100% sure how much congestion will reduce short term as a lot of things might end up causing more miles to be driven:
- people may be willing to tolerate even higher commute times in exchange for lower housing costs
- if cars self drive there is less reason for young people not to have access to them (or have more after schools activities then their families can currently drive them to)
- elderly people who are no longer safe to drive themselves can regain that sort of mobility, and I expect many of them will
- disabled people (blind, or even just folks who have poor night vision, folks with reduced motor control, and many more)
- dog delivery for lunch time walkies, why have a strange walk your dog when you could get a little light exercise? You just need a dog door with a timer and a garage (no, I doubt this will be a large number of miles, but I think this is the kind of thing in the "long tail" of "nobody does it now, but a few people will do it in the future")
- parking lots can move far from buildings (live in SF, it might cost less to have your car parked in another city overnight and come pick you up, that'll add a lot of road miles!)
- car sharing inside a family can have the car taking someone to work, driving back home, and taking other family members to work (or social activities)
- if most people drift away from individual ownership of cars and start renting time from a fleet of cars they will spend time driving from the last drop off spot to the next pick up spot
- the cost of "same day delivery" will drop a lot, and it might not all displace purchases you would have driven to a store for anyway (then again, if it is "same day", not "ASAP" shopping could be combined from multiple stores to multiple people, so maybe it could be less traffic for that)
Some other things I think might be interesting, if congestion starts to get bad again with all self driving cars you can have lot more gradations of lane limits. Rather then just "carpool" lanes that also allow low/no emissions vehicles during specific hours you could end up with surge rates where as roads get more congested you only get to use some roads/lanes/routes if you are willing to pay more. (and in a utopian psudo-libratarian future that money would pay to add lanes or roads to reduce congestion, but I think in the real world that'll be an extra tax only part of which will go to roads, and counter productively may encourage local movements NOT to do anything to fix congestions because that is how they are funding schools, police, and the governor's reelection campaign!)
Emergency vehicles are currently very hampered by congestion. The sirens can only be heard so far away, and while most people try to do the right thing, some might think the right thing is to get out of the traveling lanes so the cops/fire/ambo can use them, other might think they need to stay out of the "break down lanes" so the cops/fire/ambo can use them. Some might decide to do either of those things "in a second or two, after I take advantage of the gap!". With self-driving vehicles you will get a consistent behavior, and traffic can be alerted along an entire route as soon as needed. This could save a lot of lives.
Most traffic accidents are caused by poor driving, or drivers medical issues. They will go away. Vehicle failures will still cause accidents. They might cause fewer (self driving cars might refuse to drive if they haven't been maintained properly, and even if they don't, it is a whole lot easier to keep to a maintenance schedule if the car takes you to work, and takes itself in for service, and picks you back up after work as per normal, not waiting until you can find time for it). They might cause more (sensors add more points of failure). They might cause less (if programmed to deal with a mechanical failure, and properly tested a self driving car won't forget how to deal with correctable failures like flat tires, and even ones that have limited correction ability like break system failures self driving cards could far faster and more effectively warn other cars to get out of the way)
Accidents cause more accidents (people gawking at an accident aren't paying as much attention to driving as they should). That won't happen with self driving.