what is human "nature"?

Ignore what seems to our ear the anti-Semitic tone.  Celsus used a (fictitious?) Jew to mount various criticisms against the Christians, so Origen was responding to "Celsus' Jew."

Origen, Contra Celsum II.29
[Celsus' criticism is that] "the prophets say that the one who will come will be a great prince, lord of the whole earth and of all nations and armies."  But it is just like a Jew, I think and consistent with their bitterness, when he reviles Jesus without giving even any plausible argument, saying: "But they did not proclaim a pestilent fellow like him."  Indeed, neither the Jews nor Celsus nor anyone else could establish by argument that he was a pestilent fellow who converted so many men from the flood of evil to live according to nature (κατὰ φύσιν) practising self-control and the other virtues.

It is interesting that Origen says that Jesus converted people "from evil to live according to nature."  Jesus recovers for us and converts us to the "natural" way to live, the truly human way to live.  So is human nature evil?  I find the word "nature" in this last sense troubling, not because I don't believe that humans are wicked and have a bent toward wickedness, but because it is not a dominant term used in the NT to refer to the wicked condition of a humanity and it is not entirely theologically coherent, as I will try to briefly demonstrate below.  The only use of φύσις in the NT that comes close to affirming wickedness as human nature is Eph 2.3: "We also once conducted our lives among these [trespasses and sins] in the desires of our flesh, willing the things of the flesh and of our minds, and we were children of wrath by nature (φύσει) like the rest."  And yet, verse 2 also affirms the presence of the "ruler of the authority of the air, the spirit who presently is working in/among the sons of disobedience."  Paul, assuming he wrote the letter, still doesn't evade the presence of deception among the disobedient.  Adam and Eve rebelled AND were deceived.  This is true of our world too.  Paul uses other language to describe the wicked state of the disobedient, but his lack of "nature" talk when it comes to humanity's wicked bent might be telling.

As David Bentley Hart has said somewhere (I think in his little book, The Doors of the Sea), the East has always rejected original guilt from original sin.  But he is quick to say that this doesn't change the fact that all humans will sin.  The world was altered by the rebellion in the garden, and the result is now that all humans inevitably sin due to a messy complex of problems.  Sin makes us less human, not more so.  Even Henri Blocher, who is quite reformed in his thinking, has argued in his book on original sin that sin has become our quasi-nature.  It is a warped nature in that it has parasitically latched on to our humanity and abused our "good" human nature.  As you can see, like Blocher, I tend to affirm the classic Augustinian view of good and evil, even though I tend to depart from Augustine on original sin.

Do we sin "by nature"?  If by "nature" we mean "easily," then Yes, but we are no longer using nature in the traditional sense of those necessary properties that constitute a certain thing.  If we maintain a traditional sense of this word, then No; we do not sin by nature but παρὰ φύσιν, against or contrary to nature, which is how Paul describes homosexual behavior in Rom 1.  Anyway, I think this is important because it not only reminds us that being human is a good thing, even though humans are sinners.  It allows us to affirm the goodness of God in spite of evil. It also helps us view our life as being renewed in the image of the Son, and it validates the necessity of the virtues in the life of the Christian, which the Fathers championed so well.  Much more could be said on a number of these points, but this will suffice for now.