What's a Papist exactly?

Papist in the most simple term means follower of the pope. The reformation gave rise to groups of Christians that followed beliefs rather than individuals (e.g. only scripture, only faith, only grace). There is no need for a pope in this system of theology and those who clung to a pope were viewed by reformers at best as Luddites, heretics at worst - hence the derogatory nature of papist.

The term is a polemic and an epithet describing anyone who, in the estimation of user, is more beholden to the Pope than to Christ.

In Pilgrim's Progress, John Bunyan unfortunately expresses an opinion of the following sort: (From Section 4, The Valley of Humiliation http://www.covenantofgrace.com/pilgrims_progress_shadow_of_death.htm)

Now I saw in my dream, that at the end of this valley lay blood, bones, ashes, and mangled bodies of men, even of pilgrims that had gone this way formerly: and while I was musing what should be the reason, I espied a little before me a cave, where two giants, POPE and PAGAN, dwelt in old time, by whose power and tyranny, the men whose bones, blood, ashes, etc., lay there, were cruelly put to death. But by this place CHRISTIAN went without much danger; whereat I somewhat wondered. But I have learnt since, that PAGAN had been dead many a day; and as for the other, though he be yet alive, he is, by reason of age, and also of the many shrewd brushes that he met with in his younger days, grown so crazy and stiff in his joints, that he can now do little more than sit in his cave's mouth grinning at pilgrims as they go by, and biting his nails, because he cannot come at them.

This sort of polemic rightly earns John Bunyan the adjective "anti-Papist," and truly the term originates as an anti- thing, rather than the thing itself. Being anti-Papist means you are against the Pope, and anyone who gives him succor is your enemy. (I feel dirty even typing this, Peter, and I do apologize that this is what my church was going through at the time)

As Merriam-Webster notes, the term originates in 1535, and says;

pa•pist•ry \ˈpā-pə-strē\ noun 1535 usually disparaging : the Roman Catholic religion

In his commentary on Acts, John Calvin makes the point, saying:

f the Papists have left him any of these things, let them have the Church on their side. But if the Pope, oppressing men’s consciences with his fierce and more cruel tyranny, have disannulled and taken away Christ his government; if he have brought in a form of government altogether contrary to the gospel; if he have invented a new and strange priesthood, that he may thrust-in himself, being but a mortal man, to be the mediator between God and the world; if he have forged daily sacrifices, that he may trot them in Christ’s place; if he have invented a thousand satisfactions for sins; if he have brought reigned washings from the lake of hell, to make dry the blood of the Son of God; if he have put in his place infinite patrons; if he have torn in a thousand pieces that righteousness which must be set [sought] wholly from him; if, instead of the Holy Ghost, he have erected man’s free-will; it is, without all question, that the true Christ is banished far from Papistry. For this cause have I said that the Papists make boast of a dead carcass instead of the lively body of Christ, because, though they have extinguished the doctrine of the gospel, (which is the true soul of the Church, and which duly doth quicken the same,) yet they do greatly boast of a shadowish and trifling kind of Church.

One can see in the above quote that it is the Pope who is being villified and seen as contrary to Christ - and the Papist would be anyone who support him.

From my own reading of Luther, Calvin, etc., papist is a derogatory term referring to any Roman Catholic who accepts the Pope  as a legitimate authority from God during the time of the reformation. The reformers might not have spoken as harshly against Catholics in previous generations.  After all at some point in time even Luther would have regarded the Catholic church as having legitamate authority before turning away from the gospel of grace, to works:

Here is a sample quote from Luther clearly including all Roman Catholics under the term 'papist':

First, nobody can deny that we, as well as the papists, have received holy baptism and because of that are called Christians. (Luther's Works  41.194)