When I look at Wikipedia entries and papers online referencing Kurepa trees, the definition for a Kurepa tree includes the note that "by $\omega_1$-tree we mean a tree of power $\omega_1$ and height $\omega_1$". The height of a tree is frequently defined, but oddly enough, "power of a tree" does not seem to be.
Elsewhere in basic set theory, $\alpha$-tree means a tree of height $\alpha$ with the size of each level strictly less than the cardinality of $\alpha$. For these definitions to be compatible, it seems that "power of a tree is $\alpha$" would have a very strange definition: $\alpha$ = $\beta^+$ when there exists a level of the tree of maximum cardinality $\beta$, else $\alpha$ = the supremum of the cardinalities of all levels of the tree.
What does "power of a tree" actually mean, and where should I have looked to learn this definition?
The power of a set is its cardinality. (As opposed to its power set, which is something else.) As you noticed in the comments, Kurepa trees are supposed to have countable levels, although just saying that a tree has size and height $\omega_1$ is not enough to conclude this, so the definition you quoted is incomplete as stated.
Usually the convention is that a $\kappa$-tree is a tree of height $\kappa$, all of whose levels are of size less than $\kappa$. (This implies in particular that the tree has size $\kappa$.)
By the way, if you remove from the definition of Kurepa tree the requirement that the levels be countable, you get the notion of weak Kurepa tree, which is also referred to in the literature with the name "Canadian tree".