In the text, "Functions of a Complex Variable" The notion of infinite products were discussed which are defined in $(1.)$.
What makes an infinite product representation of holomorphic function so useful ?
What advantages do infinite products present over series ?
Finally what are the general situations that products arise in Complex Analysis ?
$(1.)$
$$\prod_{j=1}^{\infty} (1+a_{j})=(1+a_1)(1+a_2)(1+a_3)\dots$$
If we think about polynomials for a moment, we know there are two useful forms, $$ p(z) = \sum_{n=0}^N a_n z^n = A\prod_{k=1}^N (z-z_k). $$ The first is useful for things like differentiation and integration, since it is a linear combination of terms, and also good for producing things like bounds. It tells you very little about things like the locations of the roots, however. Meanwhile the product is hard to differentiate, impossible to integrate, but hands you the roots for free.
Similarly, rational functions have various forms: they are ratios of polynomials, so of course we have $$ r(z) = \frac{p(z)}{q(z)} = \frac{\sum_{n=0}^N a_n z^n}{\sum_{m=0}^M b_m z^m} = A\frac{\prod_{k=1}^N (z-z_k)}{\prod_{\ell=1}^M (z-p_{\ell})}. $$ We also have the partial fraction expansion, $$ \sum_{\ell,\alpha} \frac{A_{\ell,\alpha}}{(z-p_{\ell})^{\alpha}} + \sum_{k=0}^{N-M} B_kz^k. $$ Again, all three of these forms are useful for different things.
It would therefore be nice if the same thing happened for analytic and meromorphic functions. We know that unless $f$ is entire, the Taylor expansion of $f$ tells us only local behaviour: its high-order behaviour is dominated by the nearest singularity. Of course analytic continuation tells us that the Taylor expansion actually does contain all the information that we could ever want about the function, but it is obscured by the nearest singularity. Even if the function is entire, like sine, the convergence may be far too slow to give us useful information.
So, we would like is a factorisation of an entire function that tells us where the zeros are, and converges "better" than the Taylor series does: absolutely locally uniformly on the whole plane. Such a factorisation exists, by a theorem of Weierstrass. The reverse also holds: if we write down a random expansion of Weierstrass form, it is an analytic function. I'll give some examples where this is explicitly useful, rather than speak about general theories, since the general theories are normally motivated by understanding such examples.
The Gamma function
It is relatively easy to establish that $$ \sin{\pi z} = \pi z \prod_{n \neq 0} \left( 1-\frac{z}{n} \right)e^{z/n}, $$ and because the product converges absolutely, one may rearrange it into the familiar form that Euler hypothesised and Simply Beautiful Art alludes to in the comments, $ \pi z \prod_{n=1}^{\infty} (1-z^2/n^2) $. There is one really nice thing about these products: unlike a Taylor expansion, they don't have a "preferred" point around which they are a local approximation; since the expansion is agnostic with regard to ordering, we can get a good local approximation by choosing the terms that deviate most from $1$ nearby. Periodicity is unfortunately a casualty of the expansion: it's not that easy to prove this expansion is periodic. Indeed, the easiest way is to consider the logarithmic derivative, $$ \cot{\pi z} = \frac{1}{z} + \sum_{n \neq 0} \frac{1}{z-n}-\frac{1}{n}, $$ differentiate again, obtain a clearly periodic expansion, and then integrate back up. This formula for the cotangent is of independent interest, since it expresses it as the infinite version of a partial fraction expansion: the Mittag-Leffler expansion, which provides infinite expansions of meromorphic functions about their poles in a similar way.
One might well ask what happens when we just take half of the sine expansion: what is an analytic function with zeros at the nonpositive integers, $$ z\prod_{n=1}^{\infty} \left( 1 + \frac{z}{n} \right) e^{-z/n}? $$ It turns out through a simple calculation that this is an exponential factor away from the reciprocal of the gamma-function, $$ \Gamma(z) = \frac{e^{\gamma z}}{z}\prod_{n=1}^{\infty} \frac{n}{z+n}e^{z/n}, $$ (one can show this satisfies Wielandt's theorem, for example), which requires fairly minor massaging to relate to Gauss's definition. Then the formula $\Gamma(z)\Gamma(1-z)=\pi/\sin{\pi z}$ follows almost as a formality. One can in fact develop the entire theory of the trigonometrical functions using these ideas; this was done by Eisenstein, and a modern exposition is found in Walker's Elliptic Functions: A Constructive Approach, which is sadly now very difficult to get hold of. Speaking of elliptic functions...
Elliptic functions
Given this characterisation of sine, a natural question is what happens if we try to form a doubly-periodic function by the same principle, i.e. what is $$ \sigma(z) = z \prod_{\rho = 2n\omega_1+2m\omega_2 \neq 0} \left( 1-\frac{z}{\rho} \right) e^{z/\rho+z^2/2\rho^2} $$ where $\Im{\tau}>0$? (The extra factor in the exponential is required for convergence.) It turn out that this is almost doubly-periodic, but changes by a multiplicative factor $e^{\eta_i(z+\omega)}$ (in fact it could never by doubly-periodic: an analytic doubly-periodic function is constant by Liouville's theorem). Taking the logarithmic derivative gives $$ \zeta(z) = \sigma'(z)/\sigma(z) = \frac{1}{z} + \sum_{\rho = 2n\omega_1+2m\omega_2 \neq 0} \frac{1}{z-\rho} + \frac{1}{\rho} + \frac{z}{\rho^2}. $$ This has simple poles at each $\rho$. It is still not doubly periodic: it gains an additive factor $\zeta(\omega_i)$ when $z$ is replaced by $z+2\omega_i$. But $-\zeta'(z) = \wp(z)$, the Weierstrass P-function, is doubly periodic. So using products actually leads eventually to a doubly-periodic function, using only a simple extension of the idea of a singly-periodic function of the sort we know and love.
One can produce some quite startling number-theoretic results by using these sorts of ideas with functions similar to the $\sigma$ function, the Jacobi theta-functions: for example, the number of ways $r_k(n)$ of writing $n$ as a sum of $k$ squares is given by $$ \sum_{n=0}^{\infty} r_k(n) q^n = \vartheta_3^k(0,q), $$ where $\vartheta_3(z,q)$ is the theta-function $$ \vartheta_3(x) = \sum_{n=-\infty}^{\infty} q^{n^2} e^{2\pi i z} = G\prod_{n=1}^{\infty} (1+2q^{2n-1}\cos{2z}+q^{4n-2}), $$ where $G=\prod_{n=1}^{\infty}(1-q^{2n})$.
But perhaps the most famous infinite products are
Zeta- and $L$-functions
The Riemann zeta-function is infamously given in the half-plane $\Re(s)>1$ by the Dirichlet series $$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}. $$ The fundamental theorem of arithmetic (i.e. uniqueness of prime factorisation) implies that this is equal to the Euler product $$ \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}. $$ As mentioned above, the Euler product gives us a very useful result: the zeta function has no zeros with real part larger than $1$. This and the functional equation, relating $\zeta(s)$ to $\zeta(1-s)$, immediately tell us that we only have to worry about whether there are zeros in the strip $0 \leq \Re(s) \leq 1$, the critical strip. Of course the Riemann hypothesis is a considerable strengthening of this, that all such zeros lie on $\Re(s)=1/2$.
Similarly, given a Dirichlet character $\chi(n)$, one defines an $L$-function by the Dirichlet series $$ L_{\chi}(s) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s} \quad (\Re(s)>1) $$ Dirichlet characters are multiplicative, which means this function can also be written using the Euler-like product $$ \prod_{p \text{ prime}} \frac{1}{1-\chi(p)p^{-s}}. $$
Now, here's a mysterious thing to note: one can cook up examples of Dirichlet series that, when analytically continued, satisfy a functional equation and have zeros off the critical line $\Re(s)=1/2$. But we've never found a function that has an Euler product, a functional equation, and zeros off the critical line. Somehow one feels that even though the Euler product doesn't converge for $\Re(s) \leq 1$, its existence appears to have cause some sort of restriction on the possible locations of zeros, even outside the domain of convergence. We may find a counterexample tomorrow, of course. But it's certainly puzzling!
The importance of the locations of the zeros comes from an expression one can derive using the Weierstrass product for the zeta-function, the Hadamard product $$ \zeta(s) = \frac{(2\pi)^s e^{-(1+\gamma/2)s)}}{2(s-1)\Gamma(1+s/2)} \prod_{\rho} \left( 1-\frac{z}{\rho} \right) e^{z/\rho} $$ (indeed, Riemann wrote a formula close to this in his famous paper, which was a significant motivation for the discovery of Weierstrass's general product theorem, along with so much else). This product is used to derive the Riemann and von Mangoldt explicit formulae for the zeta function, one way of observing its relationship to prime-counting functions.
I hope that's sufficiently many examples of analytic functions with infinite products and the properties therefrom to see that they are very useful!