Probably the most common question asked by riders new to cyclocross is: "Can I use a mountain bike? Isn't it over the maximum tire width?" At local, casual, amateur races, the answer is almost always "yes" - bring whatever bike you have, come out, have fun, no worries about the size of your tires.
These riders have those concerns because they've heard that that cyclocross has a maximum tire width rule - 33mm - and they're right, it's just that those rules are typically not in place at races that aren't held under the UCI umbrella. For the most part, this means that amateur racers never have to worry about the tire width rule. EXCEPT for national championships events.
Not understanding the rules can be a serious "gotcha" and it's important to understand the details in advance. Even worse, the tire width rule, specifically in place at national championships, is enforced differently, or not at all, depending on your race category!
There are essentially three tiers - first, the "UCI" tier which applies to Elite, U23, and Junior 17-18 riders (max 33mm), "UCI Light" (35mm), for Masters categories, and "USAC" Juniors under 16 - for which there is no width restriction at all.
SHORT SUMMARY OF TIRE RULES FOR THE CYCLOCROSS NATIONAL CHAMPIONSHIP
In case you were wondering - no studded tires are allowed, regardless of category. So if the race happens to be on snow, leave the spiked/studded ice tires at home.
ADDITIONAL FACTORS TO CONSIDER
Tires are often wider than their label, and might violate the tire width rules despite their published width. So just having a tire labeled "33" or "35" isn't a guarantee of safety. This is most common when mounting tubeless tires to extra-wide rims, which can push the width of the tire slightly out of compliance.
So if you're heading to nationals, which cyclocross tires are "safe" and pass the width rule?
Which tires aren't "safe"?
Which tires are "borderline"?
The gauge itself is simple - I've seen officials using a "U" shaped wooden block (presumably with a 33mm or 35mm wide slot) in the starting grid. That's easy to approximate at home, before you choose equipment and travel.
Finally if you want to read the fine print for yourself, check out the Cyclocross equipment FAQ at USA Cycling.