Path components or connected components? Can anyone explain the difference between these two terms? 
Are they basically different names for the same thing or totally different things?
 A: 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 $X_x$.
A: There are plenty of famous examples that show that path-connectedness and connectedness do not mean the same thing. For instance, the topologist's since curve is connected but not path-connected. 
It may be useful to think of the german term for connected. It basically translates as "hanging together". This term lies much closer to the topological meaning of connectedness than 'connectedness' does (at least to me). Path connectedness is perhaps what you first think of when you hear the term 'connected'. Any path connected space must be connected, but the opposite does not hold as the topologist's since curve shows. If you use 'hanging together' instead of connected, then the theorem becomes: any space that is path connected is handing together, but not every space that is hanging together is path connected. I hope this helps.  
A: Let $(X,\mathscr{T})$ be a topological space. Two points $a,b \in X$ are path-connected if there is a continuous path $\gamma:[0,1] \to X$ with $\gamma(0) = a$ and $\gamma(1) = b$. That is, there is a continuous path from $a$ to $b$. You can check for yourself that this defines an equivalence relation, and decomposes $X$ into maximally path-connected pieces. A point in $X$ is path-connected precisely to those points in its path component.
A subset $U \subset X$ is connected if it cannot be written as the disjoint union of two open sets. A connected component is a connected subset of $X$ that cannot be made any larger and still be connected.
It is not hard to prove that if two points can be connected by a path then they lie in the same connected component. That is, a path component is connected. However, it might be the case that a path-component can be properly extended to a connected component. In this larger subset there will be a pair of points that cannot be joined by a path. Still, this subset cannot be written as the disjoint union of two open sets. This situation can arise because topological spaces can be quite strange and requiring that two points be connected by a path is a stronger notion of connectedness than often arises.
A classic example of this situation is the Topologist's sine curve. 

This is the right-hand side of the graph of $\sin(1/x)$ plus the point at the origin. It turns out this is connected. However, it is not path-connected. One cannot find a path connecting the origin to any other points on the curve. Therefore we say that the Topologist's sine curve has only one connected component but two path components, one containing the point at the origin and the other consisting of the rest of the curve.
