TAI is based upon an ensemble average of atomic clocks in national laboratories. UTC (which is used for civil timekeeping and astronomical calculations, so I would say is the primary standard). As lots of people have already said, UTC is TAI corrected to mean solar time, which itself is not all that useful as there is annual variation of the difference between mean solar time and apparant solar time of up to 16 seconds. Apparant solar time is what a sundial measures.

Rob is right that the reason we need leap seconds is that the length of the day is not exactly 84,600 seconds, but the reason for this is that the rotation rate of the earth is in fact slowing down. Sure that slowing is subject to variations due to geological effects, and as Rob says this is the reason that the insertion of leap seconds is irregular. But in the long term the earth's rotation is slowing down, as tidal friction transfers angular momentum from the rotation into the orbit of the moon. As a consequence, the moon is getting further away. The rate of the slowdown has been determined by F.R Stephenson and L.V. Morrison to be an increase of the length of the solar day of 1.7 milliseconds per century. If we continue to insert leap seconds they will get more frequent in future millenia.

And as Rob says, GPS time is not affected by any of this, it is its own atomic standard, similar to but offset from TAI. Kristin Lippincott, who despite what Steve says is an historian (of art and ancient manuscripts) seems to have strayed a little far from her area of expertise here.