Plain English Meaning
Three categories that tell you how strong an airplane's structure is and what kinds of maneuvers you're allowed to fly in it. Normal is for regular flying, utility allows some mild maneuvers, and acrobatic is built tough enough for full aerobatics.
Full Definition
The three standard airworthiness categories used to certify most general aviation aircraft, each defining the maximum load factor (g-force) the airframe is certified to withstand and the types of maneuvers permitted. Normal category aircraft are limited to non-aerobatic operations and certified to a positive load limit of +3.8g. Utility category aircraft are certified for limited aerobatic maneuvers (such as spins, if approved, , and ) at +4.4g. aircraft are certified for aerobatic without restriction other than those shown necessary by required flight tests, at +6.0g.