I'm not fully versed in SSPX, but the bottom line is they are in schism (although they, I don't think, would agree) with the Catholic Church.
I'm almost sure anything I say here will not be 100% accurate, so I encourage you to do more research if you need to know.
They do not recognize the Novus Ordo Mass as being a valid/licit Mass. They are often referred to as the "traditionalists" (think Mel Gibson's father) although I think that term is broader than the SSPX. They do not recognize Vatican II and (I'm assuming) that any pope that does cannot be a valid pope.
The SSPX priests are true priests and their masses are valid (however, I don't believe they are licit). The bishops, however, have been excommunicated. Those that participate in the SSPX masses are not automatically excommunicated, however, there may be an issue regarding obedience.
Off of EWTN, I found this:
This group began with the ordination of 4 bishops, illegally, by its Founder, Cardinal LeFebre, thus establishing a schismatic group, not an heretical one. In time many ,if not most, schismatics fall into heresy. Thus no Catholic in union with the Holy Father should be attending their services, as valid as they may be, since it encourages further splits in Catholicism. God bless. Fr. Bob Levis
Their most common "identity" is their insistance that the Traditional Latin Mass (the Mass of 1962...or the Tridentine Mass) be restored. Now it has, although the Novus Ordo is still to remain the usual missal, and they are certainly not appeased. Remember, the Tridentine was NOT fully banned. It was just really limited.
The Vatican has been meeting with them to try to bring them back to the fold. It seems not to be going as well as would be hoped for.
This is merely an overview. I'm sure there is more, and I'm sure there are more that know more than I do.