Well, part of the fitting is to determine what you already know; swing speed of various clubs. That in turn determines your position along a bell curve of player profiles that will indicate the type of clubs you should buy. Players with stronger, faster swings should use clubs with stiffer shaft flexes, lower lofts (or a wider range of lofts) and less draw bias or even a fade bias (a strong swing, properly executed, will torque the clubhead to the inside as the shaft releases its flex; however, most clubs come with a slight inside angle to the face at rest to correct for a natural tendency among amateurs to fade or slice shots).
In addition, standing there hitting balls, your ability to be consistent will lead the clubfitter to recommend clubs of various construction. Most "beginner" or "casual" clubsets are designed for maximum forgiveness; they're heavily "perimeter-weighted" for distance and maximum sweet spot, higher in mass for a particular loft, with wide, deep soles to be more forgiving of "turfed" shots and to launch the ball higher (compensating a little for the "de-lofting" of clubs in this range over time, allowing players with slower swings to hit further). These are very easy to swing, but they emphasize making a straight shot, so it can be very difficult to "shape" the ball's flight path with an intentional variation in swing.
Sets designed for more advanced players that need this ability to get out of a jam (instead of beginners who need clubs that help them avoid getting into a jam in the first place) tend to be more blade-like; narrower soles, thinner construction, etc. Still perimeter-weighted, but you'll definitely know when you're not hitting it well with a club like this.
All this knowledge will tell you the type of club to go for; then you'll choose clubs or matched sets falling in that category, and take some demos to the bay to try out. Once you've found a set that works well for you, there are additional fitting steps that further match those clubs to your swing. First and most important is shaft length. Players naturally bend over more or less at address, which affects how "thin" or "deep" their average shot is. Players who bend over a lot, and don't want to change, can shorten their club shafts to avoid "turfing" the club (hitting the ground before the ball; reduces launch speed and distance, causes inconsistent launch directions, and really pisses off groundkeeping staff). Players who stand straighter can use longer shafts to avoid "thinning", "skulling" or "topping" their shots (hitting low on the face causing a lower launch, or in the extreme hitting it with the club's leading edge or even hitting over the top sending the ball down into the ground).
Once that's sorted out, the next most important thing is lie angle, which is the angle the sole of the club makes with the shaft, and therefore its angle to the ground at impact. Simply stated, the center of the sole of each of your irons should contact the ground first. If the sole contacts the ground closer to the heel, that rotates the clubface so the ball will launch left, and it's true in reverse if the toe contacts first. To measure this, the clubfitter will place a piece of pressure-marking tape (similar to what's used in some receipt printers; at high pressure it turns from white to blue) along the sole of the club, then have you hit a few balls off of a rigid surface placed over the turf in the bay. The tape will be marked where the club hits the plank, and the fitter will bend the hosel of the club to compensate for the club hitting off-center.
Of course, all of this requires you to have a consistent enough swing, both from swing to swing and day to day, that the data produced by the fitting process is indicative of your actual swing out on the course. If your ball-striking is inconsistent or your swing path varies from shot to shot, the data from a fitting will be inconclusive. Perhaps worse, you could dial it in one way in the fitting bay, then the very next day your swing behavior will drift to the opposite extreme, so not only do you have consistency problems, your clubs will magnify them. If you can't make the same club hit the same ball in the same direction three times in a row on two different days, then take whatever clubs you have now to the driving range and hit them until you can, before you start talking about custom-fitting a set.