Generally, in the automotive world, there are two different types of dynamometer (or dyno for short) which might be used: chassis dyno; engine dyno. I'll discuss the general procedures for each type of dyno. Every dyno operator has their own procedure and will operate it how they see fit. You have to allow for this mainly because it's their dyno, they paid for it, and they have to pay to have it fixed if something goes wrong. Let them do their thing and you'll be golden. You'll find by reading this answer, though, you may want to ask questions of how they do operate their dyno to ensure you get the best (and most accurate) results possible.
As I wrote in this answer there are several types of chassis dynos. I won't rehash that here. In general, a chassis dyno is the type where you drive your car onto it to get the readings.
To get the most accurate readings from the dyno, you car should be fully warmed up to include the drive train. I've read that there can be as great as 1.5% increase in the power/torque readings between a cold drive train and one that is fully warmed. It's also a good idea as a dyno test is very stressful on an engine and vehicle, so having it warmed up only makes sense.
The first thing you'll do (after you've found they dyno you're going to use and have set the appointment) when you get to your dyno appointment is to pull up onto the dyno equipment with your drive wheels on the roller. The operator will either have you do it and guide you where to stop, or they will do it for you.
Once the vehicle has been correctly located onto the roller(s), you should then set the e-brake, turn the ignition off, and pop the hood. The operators will then set to placing tie down straps to hold the vehicle in place as well as placing chocks on both side of the non-driven tires. They will also place their electronic measuring equipment on your vehicle, such as a sensor off of your sparkplug wire to get engine RPM and a ground wire. If you've opted for a wide-band O2 (lambda) measurements for tuning, they'll get that hooked up as well. They'll also place a fan in front of the radiator to ensure a good air flow over the radiator to prevent overheating your vehicle. If your vehicle has traction control, you'll need to turn that off. More than likely, the ABS light (if your car is equipped) will come on during the run. That will go off when it's back on the road, so should be of no worries to you.
Once everything is tied down, connected, and in place, they are going to have you start your car and get it rolling on the dyno, most likely to 15-20mph. This will center the vehicle on the rollers. Then they'll check all the straps and connections to ensure your vehicle is secure. You wouldn't want your vehicle to go skyrocketing off of the dyno because it isn't! (Here's some dyno failures for you. There are plenty more out there to watch.)
Now you'll start your dyno runs. It's usually good to get (at least) three runs in to make sure the numbers you are seeing are about the same. This will ensure an accurate reading.
Most vehicles will be tested in the 1:1 (direct drive) trans gear ratio as this yields the best results. All dyno pulls (a pull is a single run) are made at Wide Open Throttle (WOT) from any beginning RPM to the redline of the engine. When any single pull is finished, the vehicle should be clutched, put in neutral, and given time for the wheels to spin down. The dyno has it's own braking system so there is no need to apply the vehicle brakes ... just let it do it's thing.
Again, these cars will be tested in 1:1 trans gear ratio. The dyno pull will begin just above the threshold where the vehicle down shifts to 2nd gear at WOT. To start the pull, you find the MPH speed which the vehicle is comfortable at, the operator starts logging data, then the vehicle is brought to WOT and is kept there until the redline. Once at the redline, the gas pedal is released and transmission is put into neutral to prevent engine braking on the dyno. Everything needs to stop spinning and then the vehicle can be put into park.
As I stated, to ensure you have the best results, there should be three runs made. This determines a good baseline for the vehicle. A typical 3 pull session should take around 15-20 minutes to complete. If you are also getting tuning done on your vehicle, you can expect it to take much longer ... most likely at least an hour if the tuner is proficient with your type of vehicle.
After all the testing is done, they'll unstrap, detach, and unhook your vehicle and you are back on the road.
If you are the average Joe-Shmoe, you'll most likely only ever use a chassis dyno. If you are really serious about engine building and getting an engine right before it ever gets into a vehicle, you may want to utilize an engine dyno to measure the power and torque of your engine directly. Strapping it to an engine dyno can also allow you to run-in or break-in an engine prior to it getting into your vehicle. This allows you to do tuning, check general operation, and find any leaks which may be present after rebuild. It is very unusual for a person to pull an engine just to check it on the dyno. A chassis dyno can do the job with the engine in the vehicle, so it would be an extra effort to do so. Plus it usually costs more to run an engine on a dyno than a vehicle on a chassis dyno.
When taking your engine to be dyno tested, you can just take your engine in to get a baseline power/torque reading, but for the most part you'd take your engine to the dyno to be tuned. I will base the rest of this answer as if the engine was going to be tuned as if you are just going to do break-in and baseline, it will be self-evident.
Obviously you will want to make sure your engine is ready to go to the dyno for testing. It needs to be complete with everything which it will need in order to run on the dyno. This would include the computer (if so equipped) or carburetor. You'll also need to bring your headers or exhaust manifolds. If you are looking to tune the engine so it will run in the car correctly and at its best, this is important. Engine dyno shops may have exhaust parts which will fit your engine, but their "dyno headers" will most likely not be what you'll be running in your vehicle, so having your own exhaust will give you better results in the end. You'll also want to bring in the type of fuel (make sure to bring more than you'll need) you'll be running in your engine, any nitrous equipment, and whatever spare parts you might have for it. It is a huge waste of time to bring an engine to a dyno just to have your time and money wasted over a $5 part. Also, don't forget your flywheel (or flexplate) and starter.
Once the engine is in the dyno facility, they it will usually get set onto some kind of cart which will allow the engine to be easily bolted to the dyno brake. It will also provide a firm foundation so the engine doesn't get loose during the pulls. The engine will then be hooked up so it will be able to run. The power, coolant, fuel, throttle, and exhaust will all get hooked up. They'll also hook up any monitoring equipment which will allow them to see how the engine is doing like a wide band O2 sensor, water temperature, and oil temperature. The fuel you brought will then be put into (most likely) a fuel cell in preparation for the dyno runs. Double check to make sure there is oil in the crank case! Put it in now if you brought the engine completely dry.
Once everything is hooked up and the engine is ready to go, the operator will start the engine and bring it up to operating temperature. Just like with the chassis dyno, the engine is going to run better and be more consistent when it's warmed up. If the engine is brand new, the operator will take the time necessary to break it in. Also if new and after break-in, the engine will be shut down and checked to ensure there's no overt fluid leaking and for general health of the engine. After that, it the engine has cooled down, it will be brought back up to temperature for the dyno session.
You'll need to tell the operator what the expected redline of your engine is so they can set an upper limit of how high to go with your engine. If you tell them 10k and the engine is only good to 6500, expect to be helping to clean up the dyno room from the shrapnel which was once your engine.
So the pulls begin. The operator will monitor how the engine is doing. They will engage the dyno to start recording the pull, then apply throttle smoothly until it get's to WOT, will watch as the RPMs increase to the preset redline, then pull back on the throttle quickly to bring the rpms down. The pull will only take seconds. You can then look at the numbers (ie: power, torque, air/fuel ratios, etc.) to see if things are as they should be. If tuning needs to be done, then now is the time to do it. Depending on how long the tuning takes, you may need to get the engine back up to temperature again before the next pull. You'll continue with the pull and tune until the engine is where it's felt to be running at its best. Then there should be three runs done on the engine to ensure your results are accurate.
Depending on how much tuning needs to be done, this could be an all day endeavor. It takes a lot of work to get the engine onto the dyno, broke-in, unmounted, and back to you.
I'll restate, these are the general procedures for getting your vehicle or engine dyno'd. Every shop can and will do things a little differently. If you are unfamiliar with a shop, call them and ask what their particulars are so you don't get surprises when you get there. Some shops will do the tuning for you, but expect to pay more for those services. The more time you vehicle/engine is strapped to a dyno, the more it's going to cost. Get all of this stuff figured out in advance and you won't be sorry.