Dominated Convergence Theorem
Consider the sequence of functions on $(0,1)$ $$f_n(x) = \begin{cases} n & \text{ if } x \in (0,1/n)\\ 0 & \text{ otherwise} \end{cases}$$ We have $\lim_{n \to \infty} f_n(x) = 0 = f(x)$. However, $$\lim_{n \to \infty} \int_0^1f_n(x)dx = 1 \neq 0 = \int_0^1 f(x) dx$$ The key in the dominated convergence theorem is that sequence of functions $f_n(x)$ must be dominated by a function $g(x)$, which is also integrable.
I would like to provide a slightly more abstract framework to illustrate this "loss of compactness"$^{[2]}$ phenomenon. The functional setting is the $L^1(\mathbb{R})$ space: \begin{equation} L^1(\mathbb{R})=\left\{f\colon\mathbb{R}\to\mathbb{R}\ :\ \int_{-\infty}^\infty \lvert f(x)\rvert\, dx<\infty \right\},\qquad \lVert f\rVert=\lVert f\rVert_{L^1}=\int_{-\infty}^\infty\lvert f(x)\rvert\, dx. \end{equation} Here we take a bounded sequence $f_n$ that converges pointwise a.e. : \begin{equation} \begin{array}{cc} \|f_n\|\le C, & f_n\to f\, \text{a.e.} \end{array} \end{equation} The question is: does this sequence converge, that is, is it true that \begin{equation} \lVert f_n-f\rVert\to 0?^{[1]\ [2]} \end{equation} This would make for a very desirable property of our functional space. However, as other answers very clearly show, the answer is negative in general. The problem is that our space is subject to the action of noncompact groups of isometries. Namely, one has the action of the translation group \begin{equation} \begin{array}{cc} \left(T_\lambda f\right)(x)=f(x-\lambda), &\lambda \in (\mathbb{R}, +) \end{array} \end{equation} and of the dilation group \begin{equation} \begin{array}{cc} \left(D_\lambda f\right)(x)=\frac{1}{\lambda}f\left(\frac{x}{\lambda}\right), &\lambda \in (\mathbb{R_{>0}}, \cdot) \end{array} \end{equation} The change of variable formula for integrals immediately shows that those group actions are isometric, that is, they preserve the norm.
So, fixing a non-vanishing function $f\in L^1(\mathbb{R})$, its orbits $T_\lambda f$ and $D_\lambda f$ form bounded and non-compact subsets of $L^1(\mathbb{R})$. In particular, letting $\lambda \to +\infty$ (or $\lambda \to -\infty$ for translations, or $\lambda\to 0$ for dilations), one finds counterexamples to the question above. (Note that, more or less, all the examples constructed in the other, excellent, answers are constructed this way).
In technical jargon one says that the translation and dilation groups introduce a defect of compactness in $L^1(\mathbb{R})$ space. This is the terminology of the Concentration-Compactness theory (the linked page is a blog entry of T. Tao, but the theory has been founded by P.L. Lions). The dominated convergence theorem can be therefore seen as a device that impedes the defect of compactness to take place.
Footnotes
$^{[1]}$ The OP only asks about the convergence of the integrals: $\int f_n\to \int f$. Now a standard theorem (cfr. Lieb & Loss Analysis, 2nd ed. Theorem 1.9 (Missing term in Fatou's lemma), see in the remarks) gives us the equivalence \begin{equation} \begin{array}{ccc} \lVert f_n\rVert_{L^1} \to \lVert f\rVert_{L^1} & \iff & \lVert f_n-f\rVert_{L^1}, \end{array} \end{equation} Therefore, at least for sequence of positive functions for which $\int f_n=\lVert f_n\rVert_{L^1}$, the failure of convergence for sequences of integrals is exactly the same thing as the failure of convergence in $L^1$ space. That's why one can see the phenomenon in this functional analytic setting.
$^{[2]}$ Compactness usually means that bounded sequences have convergent subsequences. In $L^1(\mathbb{R})$ space, pointwise convergent sequences are compact if and only if they are norm convergent.