The term "+p" refers to ammunition which is loaded by the manufacturer to a pressure level above the normal SAAMI specification.   But "+p" is still a recognized SAAMI specification (e.g., 9mm+p, .45ACP+p). 

The term +p+, used only in reference to 9mm ammunition usually, refers to a round loaded beyond any SAAMI specifications.   Most NATO-spec ammunition tends to be +p+.  There is no SAAMI specification for "+p+" ammunition.

By manufacturing ammunition to a higher pressure level, companies can achieve greater velocity with any given bullet.   This means more momentum and muzzle energy, but usually an increase in recoil and muzzle flip as well (particularly in .45ACP +p).

There is no such thing as .40S&W +p!  Period.  No exceptions.    There is not a SAAMI specification for .40S&W loads beyond the standard pressure.  Any company which sells ammunition loaded above the standard SAAMI specification is selling dangerous ammunition.  There are no firearms rated to use such ammunition.  If you really feel you need a .40S&W +p, you can go out and buy a proper gun and ammunition for it.  Just go to the gun shop and ask for a 10mm.

If you are considering using .40S&W ammunition which is rated "+p" by the manufacturer, at the very least do yourself a favor and call the company to find out if they are (a) mislabeling their product for marketing purposes, (b) mislabeling their product because they do not know or understand SAAMI terminology, or (c) making out-of-spec ammunition.

The same is true for .380ACP and 357SIG and 10mm.  There is no +p specification, and ammunition which is loaded beyond SAAMI specification is dangerous.    Period.

The 9mm and .45ACP, however, does have an official SAAMI +p rating.  Most modern nines and forty-fives are rated for +p ammunition, though most will show accelerated wear if they fire a steady diet of nothing but +p ammunition.

In 9mm, most people consider only the +p or +p+ loads as viable for personal defense.  While I think there are some exceptions (the 147gr Remington Golden Saber, for example), as a general rule I'd rather have the added velocity from a +p or +p+ round.    Unlike the .45ACP +p, 9mm +p does not tend to have significantly greater recoil.

A +p load in a .45ACP handgun comes close to matching a full power 10mm in both power and recoil.  Remington produces both a normal JHP and a Golden Saber BJHP in 185gr +p moving at 1,140fps, an improvement of about 125fps over the normal pressure loads.    The Golden Saber in particular is a good design for the increased velocity.    For heavier bullets, ProLoad makes both a 200gr +p and 230gr +p using the Speer Gold Dot at 1,000 and 920fps, respectively.