Russel's paradox arises if you consider the set $U=\left\{x:x\not\in x\right\}$. Ask yourself if $U\in U$. If you suppose so, then by the definition of unrestricted set comprehension $U\not\in U$. You have a contradiction, so it must be the opposite of what you supposed, that is, $U\not\in U$. But this is the same as saying $U$ belongs to the complement of itself, that is, $U\in U$. You now have another contradiction, but this is far worse, since you have no hypotheses. The whole theory is logically inconsistent.
In set theory there are two ways for getting rid of the Russel's paradox: either you disallow the set of all sets and other similar sets (see for example the Zermelo-Fraenkel set theory), or you allow them, but you also restrict the way they are used (see for example the Morse-Kelley set theory).
In the first case, set comprehension says if you have a set $A$ you can have $\left\{x\in A:\phi\left(x\right)\right\}$ (notice: writing $\left\{x:\phi\left(x\right)\right\}$ is just wrong in this case, because you have to have an initial set). If you now define $U=\left\{x\in A:x\not\in x\right\}$ and you repeat the same passages as before, it only follows that $U\not\in A$. There's no contradiction and the theory is consistent.
In the second case, you consider classes, not just sets. Sets are classes that belong to some other class, while proper classes are classes that belong to no class. Set comprehension, in this case, says you can have $\left\{x:\phi\left(x\right)\right\}$, but all its members are sets by definition. If try to reproduce Russel's paradox, you get that $U\not\in U$. If you then suppose that $U$ is a set, then you have a contradiction, so $U$ must be a proper class. This is all you get. No contradictions. The theory is consistent.
