+P, or "Plus P", stands for "increased pressure". It denotes a type of ammo as having more pressure than the standard load for a given cartridge. That standard is commonly called "SAAMI Spec". S.A.A.M.I stands for Sporting Arms and Ammunition Manufacturers' Institute, and they were formed and charged with cataloging and detailing firearms cartridges and chamberings in 1926 with the aim of creating standards for manufacturers to follow.
As materials improve and pursuits and games evolve, so have some cartridges. A great example of this is 45 Colt, developed and in constant use since 1872. The S.A.A.M.I. specifications for 45 Colt detail a specific chamber pressure of approximately 10,000 to 18,000 P.S.I.. That data is based on the original loads from the 1800s using blackpowder. Today there are 45 Colt loads that exceed 40,000 PSI, and that level of power requires the gun to be made to handle it. Early and antique firearms chambered in 45 Colt may dangerously explode if you fired a modern, +P load. This is why ammunition that exceeds the SAAMI specs are labelled as +P. You should check the manual for your firearm before using any amount of +P ammunition in it. You should also take extra care that old and antique arms are not fired with unknown ammunition.
That said, +P ammo exists for a reason. For example, it is known among ballisticians that defensive 45 ACP loads fired out of short-barreled, compact pistols doesn't have enough velocity to expand many hollow-point bullets. The loss of velocity due to the shorter barrel, as the cartridge was designed around a longer one, is the issue. The increased velocity of modern +P and +P+ (yes, that'a thing!) loads solve this problem.