If $X$ is a space and "P" is an adjective that can apply to spaces, then in many cases "$X$ is locally P" means "for every point $x\in X$, for every neighborhood $N$ of $x$, there is a neighborhood $N'$ of $x$ in $X$ such that $N'\subset N$ and $N$ is P", or more briefly "every point of $X$ has arbitrarily small P neighborhoods". I would agree with Terry, though, in cautioning against a strict rule.
This gives the usual notion of locally connected (which turns out to be equivalent to saying that every component of every open set is open), and likewise for locally path-connected. It also gives one of the usual notions of locally compact, generally considered better than the other usual notion, which happens to agree with it in the Hausdorff case so that the difference is often irrelevant.
Here is one way in which people confuse each other with "locally" and "local". What do I mean if I say that a map $f:X\to Y$ is locally a homeomorphism? Surely I do not mean that every $x$ has arbitrarily small neighborhoods $N$ such the the restriction to $N$ is a homeomorphism $N\to Y$. What I mean is that every $x$ has arbitrarily small neighborhoods $N$ such the the restriction to $N$ gives a homeomorphism $N\to f(N)$. OK, so this conforms to the general rule of my first paragraph [extended to the case where "P" is allowed to be "a homeomorphism" even though that's not an adjective] if I give "homeomorphism" the old-fashioned meaning of "homeomorphism to its image". (Also I could have omitted "arbitrarily small", but that's beside the point.)
But that's not the confusion that I was thinking of. Here it comes. I have known more than one student to misinterpret the expression "locally homeomorphic" in the following way: you learn what a local homeomorphism is (perhaps while studying covering spaces), and then later someone says something about $X$ being locally homeomorphic to $Y$. And you assume that this means that there is a map $f:X\to Y$ that is a local homeomorphism, whereas you should have been thinking "every $x$ has arbitrarily small neighborhoods $N$ such that $N$ is homeomorphic to $Y$".*
*or more likely such that $N$ is homeomorphic to a neighborhood of a point in $Y$. See, that's an example of what Terry is saying.