TITLE: Path components or connected components?
QUESTION [6 upvotes]: Can anyone explain the difference between these two terms? 
Are they basically different names for the same thing or totally different things?

REPLY [5 votes]: Path components and connected components are not the same thing. As an extreme example, consider the unit square torus $T = \mathbf{R}^2/\mathbf{Z}^2 \simeq S^1 \times S^1$, and let $X$ be the image of a line of irrational slope through the origin. The complement $X' = T \setminus X$ is connected, but has uncountably many path components.
Clearly $X$ is path-connected, as is each translate $X_{x} = X + (x, 0)$ of $X$, and the union of the translates of $X$ is $T$. The translates of $X$ that are disjoint from $X$ are the path components of $X'$.
(On a tangent, there are uncountably many distinct translates, since $X$ hits each circle $\{x\} \times S^1$ countably many times, but each circle contains uncountably many points. Picking a set of distinct translates of $X$ whose union is $T$ amounts to the standard construction of a non-measurable set.)
To see $X'$ is connected, note that each translate $X_x$ is dense in $T$, so if $U$ is a non-empty open subset of $T$, then $X_x \cap U$ is non-empty. This implies $X'$ has no separation: If $U$ and $V$ were disjoint, non-empty open sets whose union is $X'$, then $U \cap X_x$ and $V\cap X_x$ would separate the path-connected set $X_x$.