A sponsor co-op apartment is one that is owned by a building's original owner or the corporation responsible converting the building from a rental into a co-op.
The big advantage of a sponsor co-op--and the reason they are so sought-after--is that buyers do not have to be approved by the co-op board. You may even get to bypass the building's normal downpayment and reserve requirements, and be grandfathered in on certain perks held by the sponsor such as rights to a storage unit or to install a washer-dryer.
Sponsor apartments tend to cost about 5-10% more than comparable non-sponsor units, but for many people who might not be approved by a co-op board (such as freelancers, foreign citizens, and the unemployed or retired) the alternative is buying a condo, which is much more expensive on average than a co-op.
Closing costs are higher for sponsor units, as transfer taxes (1.825% of purchase price for properties over $500,000, and 1.425% for properties under $500,000) are paid by buyers. Also beware of the "sponsor renovation" -- often a low-cost quickie renovation meant to spur a quick, high sale and emphasizing surface appeal over quality and longevity.