Is fibre product of varieties irreducible (integral)? 
Let $k$ be an algebraically closed field and $X,Y$ varieties (i.e. integral, separated schemes of finite type over $k$). Is the fibre product $X \times_k Y$ necessary irreducible or integral?  

I have another more or less related question: 

If $A,B$ are finitely generated $k$-algebras which are also integral domains, is $A \otimes_k B$ an integral domain?

 A: As mentioned by Georges Elencwajg, the fiber product is also integral if $k$ is algebraically closed. In general, some author require a variety over any field $k$ should be geometrically integral. In this way the fiber product of varieties is also variety. As for the affine case, I just want to mention an example that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}$ is not a domain. In fact it is isomorphic to $\mathbb{C} \times \mathbb{C}$.
A: a) Yes, the product $X\times_k Y$ of two varieties over an algebraically closed  field $k$ is a variety. To prove it, reduce to affine varieties and for those  use:
b)   If the field $k$ is algebraically closed and if $A$ and $B$ are $k$-algebras without zero divisors, then their tensor product  $A\otimes_kB$ also has no zero divisors.
 (Whether $A$ or $B$ is finitely generated is  irrelevant.)
This is proved in Iitaka's Algebraic Geometry, page 97 (Lemma 1.54)    
Edit: And if $k$ is not algebraically closed...
... all is not lost!    
Let $k$ be a field and $R$ be a $k$-algebra which is a domain and  with fraction field $K$.
If the extension $k\to K$ is separable and if $k$ is algebrically closed in $K$, then for all $k$-algebras $S$ which are domains, the $k$-algebra $R\otimes_k S$ will be  a domain.
(Beware that separable means universally reduced. If $k\to K$ is algebraic, separability coincides with the usual notion.)
With luck, one of $A$ or $B$ may play the role of $R$ and $A\otimes_k B$ will be a domain.  
As an illustration,  any intermediate ring  $k\subset R\subset k(T_1,\cdots,T_n)$ (where the $T_i$'s are indeterminates) satisfies the conditions above and will remain a  domain when tensorized with a domain.   
Bibliography
For the product of algebraic varieties I recommend Chapter 4 of Milne's online notes.
For the tensor product of fields, you might look at Bourbaki's Algebra, Chapter V, §17.  
New Edit
At Li's request I'll show that $X\times Y$ is irreducible.
Let $$X\times Y=F_1\cup F_2$$ with $F_i$ closed and consider the sets $X_i=\lbrace x\in X\mid \lbrace x \rbrace\times Y\subset F_i\rbrace$.
Since $$\lbrace x \rbrace\times Y=[(\lbrace x \rbrace\times Y)\cap F_1]\cup  [(\lbrace x \rbrace\times Y)\cap F_2]$$ we see, by irreducibility of $\lbrace x \rbrace\times Y$ (isomorphic to $Y$),  that each vertical fiber $\lbrace x \rbrace\times Y$ is completely included in (at least) one of the $F_i$'s. In other words, $$X=X_1 \cup X_2.$$   It suffices now  to show that the $X_i$'s are closed, because by irreducibility of $X$ we will then have $X_1=X$ (say) and thus $X\times Y=F_1$: this will prove that indeed $X\times Y$ is irreducible.
Closedness of $X_i$ is proved as follows:
Choose a point $y_0\in Y$. The intersection $(X\times \lbrace y_0 \rbrace)\cap F_i$ is closed in $X\times \lbrace y_0 \rbrace$ and is sent to $X_i$ by the isomorphism $X\times \lbrace y_0 \rbrace \stackrel {\cong}{\to}X$. Hence $X_i\subset X$ is closed. qed.
