Let's look at an example. Consider the sequence of functions
$$
f_n(x) = \begin{cases}
|x|-n &\text{if }x \in (-\infty,-n)\cup(n,\infty)\\
0 &\text{if }x \in [-n,n].
\end{cases}
$$
Graph these functions. Without thinking about the definitions you have of different types of convergence, do you think that $f_n$ converges to some other function? If you said "yes", then what's its limit? $f(x) = 0$? But for each $M \geq 0$ notice that, for each $n$, I can find an $x \in\mathbb{R}$ for which $|f_n(x) - f(x)| > M$. (Do you see an $x$ which works? Try $x = 2M + n$.) Doesn't that seem a bit weird, if $\{f_n\}$ converges to $f$? Indeed, this means that I can define a sequence $\{x_n\}$ of real numbers such that $|f_n(x_n) - f(x) > M$ for all $n$, and therefore $\lim_{n\to\infty}f_n(x_n) \neq f(x)$ (if this limit exists at all). (Worse still, you can choose $x_n$ in such a way that $\lim_{n\to\infty}f(x_n) = \infty$. I'll leave this up to you.)
Notice that $f_n \to f$ pointwise. (Prove this.)
If we can agree that this is undesirable behavior to have for convergent sequences of functions, then you already know one of many reasons why we need a stronger definition of convergence $f_n \to f$: pointwise convergence is too weak a property to be useful when discussing many properties of function sequences.
This is where uniform convergence comes into the picture. Can you see that $\{f_n\}$ does not converge to $f$ uniformly? (Try proving this from the definition.) Intuitively, uniform convergence encodes the idea that the entire function $f_n$ converges to $f$ at the same time (or same rate), not just that $f_n(x)$ moves towards $f(x)$ eventually at each point $x$. It's a global property of the functions $f_n$ and $f$ (i.e., a statement about $f_n$ involving the behavior of $f_n$ on its entire domain at once), not a local property like piecewise convergence (i.e., a property about $f_n$ that is equivalent to statements that individually only take into account small portions of the domain of $f_n$ (in this case, individuals points)). Indeed, a good exercise for you would be to show that if $f_n, f: \mathbb{R} \to \mathbb{C}$ for each $n$, then $f_n \to f$ uniformly if and only if $\|f_n - f\|_\infty \to 0$ as $n \to \infty$, where $\|\cdot\|_\infty$ is the sup-norm defined for $g:D\to \mathbb{C}$ by
$$
\|g\|_\infty = \sup_{x \in D}|g(x)|
$$
For $\{f_n\}$ and $f$ as in the above example, you can, in fact, show that $\|f_n - f\|_\infty = \infty$ for all $n$ (see the parenthetical about $\lim_{n\to\infty}f(x_n)$ above), which gives you a second way to prove that $f_n$ does not converge uniformly to $f$.
Hopefully this clarifies the concepts some. I'd highly recommend trying different examples (both from a textbook and ones you construct on your own). Conveniently, good exercises aren't to terrible to come up with on your own for this concept; whenever you know $f_n \to f$ pointwise, you can always ask whether or not $f_n \to f$ uniformly, as well.