Hmmm, well "Christian" and "Magic" are two words that would not be used in conjunction in the 13th century, but I'll give it a shot. I loved studying this era in school.
First up, I'd say look through material written for
Ars Magica; if you can find the old supplement
Maleficium, you are pretty well set for ideas on what demons can do (I can't recommend
Pax Dei, the Divine supplement, as it is probably the only pro-Divine, anti-Church-and-anything-even-vaguely-connected-with-it supplement in existence). Anyway,
Maleficium shows up on eBay all the time.
As to what people thought prayers, miracles, demons, and angels could actually do...
I would suggest looking at some medieval Saints lives for ideas of what miracles and prayers were supposed to be able to do. Most of it is curative or calming, though there is also the ability to smite evil doers (essentially anyone non-Christian, including heretics). Another power commonly found was finding lost items or discovering who stole items. Food shows up a lot.
Of course there are other, odder miracles. Stigmata (...not of the eye...) commonly manifest as a way of showing holiness, as does the notion that, when you die, your corpse smells of flowers or of apples.
Angels are a bit tricky, too. Angels guide, give wisdom, comfort, smite, and all the rest, but they cannot be commanded. Instead they aid an individual at the behest of God (or Christ or Mary or an equivalent force).
Demons would have the ability to get you to do things you might otherwise not wish to, but also have you stand by your decision after the fact. A demon might convince that, since you are mad at your sister over something minor, you must strike her in church, or drive her out of her house. In other words, demons don't force you to do things utterly outside your desires, but cause you to overreact. After the act they will make you feel foolish and leave you in a position of feeling confused, angry, and yet determined to take credit for your action (maybe with some form of justification). And demons are terribly, terribly incremental in their actions. Since they don't worry about
when you die, they are willing to go for incredibly long-range plans.
Magic, conversely, is
EVIL. Well, mostly. Usually. Unless it is done for a really, really good cause and doesn't hurt anyone and the person can't be defined as a witch, which itself is a highly variable term, mumble, mumble, equivocate... Basically, you're not going to find anyone Lobbing Fireballs For God. Magic in the 13th century was a lot of calling up the dead to find buried treasure, having forests appear suddenly, losing someone on a path, and creating wondrous items such as cloaks covered in poison or that would burts into flame when worn. The essential definitions are, if you are calling up the dead or causing someone to die, your magic is evil. If it is anything else, see your priest first and, if he approves, Handle With Care, don't use more often than directed, and go to confession immediately afterwards.
Of course if you want to make the world more fantastical, it would probably be better to base your campaign on legends written at this time, such as the early Arthurian legends, or those of Charlemagne. In these the magic is a bit more common, there are fewer problems with it, sometimes Saints and Angels come down to aid a hero at the hero's request, and magical swords appear nearly as often as in D&D.