What's the definition of a "local property"?

Not only should the property be true for a neighborhood of each point.
It must also be the case that having a neighborhood with the given property around each point implies that the entire space satisfies the property (satisfying a property locally is not the same as the property being local).

For example, being open is a local property (by which I mean that whether a subspace is open can be checked around each point of said subspace).

To add to the existing answers, be careful about two different terminologies:

• a property $P$ is local when, for all spaces $X$, if $\{U_i\}$ is an open cover of $X$ and all the $U_i$ have $P$, then $X$ has $P$. (cf. also Tobias Kildetoft for an equivalent characterization)

• if $P$ is some property of topological spaces, a space $X$ is said to be locally $P$ if every point of $X$ has a neighborhood basis of subspaces satisfying $P$.

So for example the property $P$ of being connected is not local, but you can talk about a space being locally connected, which is completely unrelated to the connectedness of the space.