It is a well known fact that path connectedness of a space is a stronger property than connectedness.
In case of a connected locally path connected spaces, these two notions are equivalent.
My question is: are there any other additional conditions that can be put to a connected space to claim that it must be path connected?
In particular, does a normal connected space have to be path connected? Is there a path connected space which is not normal?
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When $X$ is locally connected is when you start getting positive results.
Especially every locally connected, locally compact, separable metric space is locally path connected. So in particular, if it's connected then it will be path connected.
Every locally compact, connected metric space is separable, so define a generalized Peano continuum to be a locally compact, locally connected, connected metric space.
Every generalized Peano continuum is path-connected.
See Whyburn's "Analytic Topology" for this theorem.
You may also like to know that any path-connected Hausdorff space is arc-connected, i.e. for every pair of distinct points there is an embedded copy of $[0,1]$ with those points as its end points. This is also proven in that book; these results are in chapter 2. In fact, generalized Peano continua are locally arc-connected (in the strong sense; each point has an open neigborhood base of arc-connected sets).
If $X$ is compact then even stronger local properties hold, see Nadler's 'Continuum Theory' chapter 8, especially the exercises.
No, a normal connected space doesn't have to be path connected. Take, for instance, $X=[0,1]^2$ and consider on it the order topology induced from the lexicographical order ($(x,y)<(u,v)$ is $x<u$ or $x=u$ and $y<v$). It is a normal connected space which is not path-connected.
And if $X$ is a set with more than one point, if $p\in X$, and if you define the topolgy $\tau$ on $X$ which consists of the empty set and of those subsets of $X$ to which $p$ belongs, then $(X,\tau)$ is path-connected, but it is not normal