Just for the record -- "cap" is not an acronym. In other words, there's no need to capitalize the letters. CAPE is capitalized by it is an acronym (convective available potential energy). The "cap" equivalent of CAPE is CINH or CIN (convective inhibitition; "CI" is usually used to refer to convective initiation). In this regard, the following analog may be appropriate -- CAPE:cap :: CINH:LI ... I see the term "CAP" written many places, so I wanted to mention that.
As you noted, the cap (or capping inversion) tends to refer to an inversion above the convective boundary layer. In some cases, this technically may not be an "inversion" -- an isothermal layer atop the boundary layer can effectively "cap" a surface parcel as well. For that matter, any stable layer immediately above the boundary layer can "cap" a surface parcel (and there typically is a stable layer immediately above the boundary layer -- the change is lapse rate is actually what helps define the top of the boundary layer). At any rate, "cap", when used as a measurand or index, is the maximum difference between the temperature of a parcel and the temperature of the environment when the parcel is stable relative to the environment. As I noted in the analog above, you can think of the "cap" (in terms of using the cap temperature) similar to the Lifted Index (except the level of maximum capping will not be at the same height in every sounding).
There are two primary ways by which to remove the cap -- you either warm the surface or cool the capping layer. The former is typically done by diabatic heating (i.e. sunshine), and the latter is typically done by large-scale ascent or persistent low-level convergence. Of course, you don't necessarily want to remove completely the cap. In this regard, we're talking about the balance between capping and forcing. If you have only weak low-level convergence, a weaker (or non-existent) cap may be required in order to allow for surface-based convection; if there is strong convergence / forcing, a surface-based convection may occur even in areas where a stronger cap persists.
Let's look at a particular sounding:
Above, the maximum capping occurs near 750mb. Ignoring the Tv correction for a moment (i.e. use the left-most parcel / salmon-color / trace for a moment), the "cap", in terms of an index, is about 5-6C. In other words, the environment is approx 6C warmer than the parcel would be around 750mb. In this case, a tornadic supercell passed 30-40 miles N of the launch-site of this sounding, but no appreciable surface-based convection occurred nearer or south of this sounding site, despite a dryline located west of the location. Why? Well, of course, there are myriad reasons why deep moist convection does or does not occur, but the cap likely prohibited surface-based developed closed to and south of this location. It's worth noting that the virtual temperature (Tv) correction includes the effects of moisture on density... Moist air at a given pressure and temperature will be less dense than dry air at the same temperature and pressure. Tv is basically the temperature at which a parcel would need to be to have the same density as a completely dry parcel. In the case of the sounding above, there is a substantial deep moist layer... The high moisture content of the boundary layer is equivalent (from a density standpoint) to the boundary layer having a greater temperature... For this reason, you'll see a 2nd parcel trace (the right-most dashed, salmon-colored curve) that shows that the cap is weaker than it first appeared.
Similar to meteorologists preferring the use of CAPE to the Lifted Index, many folks prefer to use CINH in place of a sort of "cap index" (max temp difference between stable layer and parcel). CINH is the area between the parcel trace and the environmental profile; you can think of CINH as being similar to, but opposite of, CAPE.
In terms of "breaking the cap"... In many situations, the magnitude of CINH should be less than 50 j/kg in order to see surface-based convection; CINH of -125 j/kg may well be prohibitive of convection in all cases except those that feature very strong low-level convergence. It's tough to give a "magic number" for cap strength, as we know, since initiation, again, is a balance between cap strength (inhibition) and forcing strength. If a cold front is plowing down the Plains, surface-based convection may occur despite a strong cap (-100 j/kg CINH or 6C 'cap index'). In many dryline cases, surface-based convergence, anecdotally at least, tends to be weaker, so weaker CINH is often preferred to allow for initiation. Oftentimes, I prefer to see at least some CINH (i.e. at least a weak cap), since it may help convection from initiating over a large area (the typical "cow fart initiation" days).
