Not much. Here's our primary source, Bede:
Eosturmonath has a name which is now translated "Paschal month", and which was once called after a goddess of theirs named Eostre, in whose honour feasts were celebrated in that month. Now they designate that Paschal season by her name, calling the joys of the new rite by the time-honoured name of the old observance.
That's all he has to say. We can only speculate beyond that, and even on that much it's less than clear - unless Eostre was some kind of minor or local goddess, we'd expect to find some mention somewhere of either her or a non-Anglo-Saxon equivalent. It's entirely possible the name was the name of a Pagan festival, and not of a goddess at all. This is the position taken by Sermon^1 among others; I find this side of the argument highly compelling. The position that one mention in Bede establishes firmly that there was such a goddess does not persuade me, though there are certainly scholars who take that view (Rudolf Simek among them).
If we take Bede at face value, then, during the lunar month containing the full moon following the vernal equinox, feasts were celebrated in celebration of the goddess Eostre. That's what we know, and that's really all we have, and we don't even quite know that much for certain.
Sermon argues further that the festival was a local one, with the German name for Easter being a borrowing from the Anglo-Saxon as a result of missionaries such as St. Boniface. Most scholars instead reconstruct the two terms as cognate, inherited from a common origin (Grimm gave this as *ostara, but the modern reconstruction is *austron), which if correct suggests the holiday was widespread in the Germanic-speaking world and that we should therefore be even more surprised by the lack of attestations of the theonym if it was one.
As for what the festival was like, we have a small hint right in the name. "Eostre" is quite clearly derived from a proto-Germanic root referring to dawn and related to the word for "east"^2 which can be traced back to a proto-Indo-European root that may well have been used as a theonym even in that period. PIE *h2ausos seems likely to have been the basis for the Norse goddess Freyja and her Anglo-Saxon cognate Freo (the Germanic names come from an epithet, meaning "lady"); if Eostre wasn't a distinct goddess, the festival by that name may well have been dedicated to Freo. The fact that there's already a well-established fertility goddess tied to the dawn whose holiday might have logically been known as Eostre is both a hint at what this entailed, and a hint that perhaps Bede is mistaken about Eostre having been a theonym (given that he's a Christian and not describing contemporary practice). (It's also conceivable that Eostre was another name for Freo. I haven't seen anyone advance that thesis, probably because we lack any positive evidence on which to do so.)
Regardless, the festival may have involved imagery related to the dawn, and was likely centered around some specific goddess. We also know quite a bit about Anglo-Saxon religious practices in general that would certainly apply to how Eostre was celebrated, and feasting would be both expected and apparently confirmed by the reference in Bede.