This can happen as soon as next day after the auction. For example, with the 
2002 BMW 325ci the first notification of delivery I had was the tow truck driver calling me to tell me he was outside the house asking where I wanted the car. That was 8am the day after the auction. By the time I got back, the vehicle was delivered in the garage and he was gone.
You're going to be surprised - it will not be in the same condition as in the pictures. Just about anything can happen to it from a punctured door to a window left open leading to water damage. See the Forklifts and Transport Damage section for more details.
With all that said, it's still a better experience than buying from a dealer. Replace the shock and mild disappointment of the vehicle delivered with the manipulative, known-immoral and 'lets get the most out of you as possible' experience of buying from a dealer.
Potentially cheaper and less hassle: picking it up yourself.
If you have the equipment and know-how, driving down to the auction house and trailering the vehicle back home might be a good option. This is like the lake boat-launch experience, but with dudes trying to get paid as well. Make sure you know what you're doing to not interfere with the guys who are down there trying to pickup for their day job. The author preferred to pay the entirely reasonable $85-$125 typical delivery fee to have a professional with all the gear deliver it into their garage.
Accepting or rejecting the delivery
You don't have much choice here. Lets assume the tow truck driver damages the vehicle. You can argue with them, you can try to get them to pay to fix it, but you're going to have to accept initial delivery anyways. There's really no recourse for you - they can't take it back to the auction house. This is frequently an owner-operator of a small tow truck company, or just some random guy doing his job. This is one of a dozen or so wrecks he will deal with that day. If you make it miserable, they'll just take it back to their shop or their front yard and keep it for scrap.
The easy delivery
Tow truck driver rolls it gently off the flatbed, starts it up, and drives it into your garage. Everybody is stoked.
The hard delivery
The vehicle can't be started even though you bought it run and drive. Struggle with that for 30 minutes or so, then they reload it and then deposit it off the flatbed into your garage. Bit of a hassle, takes a long time and disturbs the neighbors with the heavy truck engine noise. A couple of trim pieces and a front fender get ripped off. Still, 30 mins later you're all set.
The Nightmare delivery
Delivery driver unloads it in the street in front of your house. He's gone by the time you get back. You can't start it, you can't push it up your driveway. Option 1 is call another tow truck company have them pick it up and then put it in your garage. If that won't work, you can get a bunch of neighborhood teenagers to push it while you drive. What if the tires are flat and won't hold air? Grab some Wheel lift dollies from harbor freight, a snatch block and a winch. Now you can pull it with the push of a button in your garage. You just have to spend about $1000 extra.
What should you have in your garage?
Non-negotiables: eye protection, fire extinguisher, and ventilation to turn over the air in the garage quickly. Everything else is highly dependent on your skillset. You don't know what's in that car, and what gases and liquids may be emanating from it. Make sure you're ready with a very large fan and personal protective equipment before you receive it.
What do you do with a new vehicle?
Make sure you do what needs to be done first to maintain the safety and cleaniness of your garage. Follow the new delivery checklist to verify it's not leaking and not stinking. Once it's ready, then put it up on your lift and start working on it.
How can you work on these safely?
Check out the safety equipment that works.
What tools do you need to make all of this easier?
From safety gear that actually works, to tool storage, lighting and cleanup. Check out the one highly opinionated guide on how to build a home garage for working on cars.