A direct set is a partially ordered set $(I,\leq)$ s.t. $\forall a,b\in I,\;\;\exists c\in I$ with $a\le c$ and $b\le c$.
An inverse system is a family of topological spaces indexed by $I$, $\{X_i\}_{i\in I}$ together with a family of continous maps $\varphi_{ij}:X_j\to X_i,\;j\ge i$ s.t. $\varphi_{ii}=id_{X_i}$ and $\varphi_{ij}\circ\varphi_{jk}=\varphi_{ik}$.
A family of compatible maps is, given a topological space $Y$, a family of continous maps $\{\psi_i:Y\to X_i\}_{i\in I}$ s.t. $\psi_i=\varphi_{ij}\circ\psi_j,\;\;\forall j\geq i$.
We define the inverse limit of the inverse system $\{X_i,\varphi_{ij}\}_{j\ge i,\;i,j\in I}$ as a topological space $X$ together with a family of compatible maps $\{\varphi_i:X\to X_i\}_{i\in I}$ s.t. the following universal property holds: for all family of compatible maps $\{\psi_i:Y\to X_i\}_{i\in I}$ there exists a unique continous map $\gamma:Y\to X$ s.t. $\psi_i=\varphi_i\circ\gamma,\;\;\forall i\in I$.
Ok, we know that the inverse limit of an inverse system exists, and up to isomorphism it's unique.
Then considering $$ C=\prod_{i\in I}X_i $$ endowed with the product topology and defining $$ X=\{c\in C\;:\;\varphi_{ij}\circ\pi_j(c)=\pi_i(c)\;\;\forall j\ge i\} $$ where $\pi_i:C\to X_i$ is the usual projection, and considering $\varphi_i:=(\pi_ {i})_{|X}$, we have that $\{X,\varphi_i\}_{i\in I}$ is the inverse limit of the inverse system $\{X_i,\varphi_{ij}\}$.
All these facts are well known.
Now, on one side up to isomorphism, the inverse limit is unique... but on the other side my book (Wilson, Profinite Groups) sometimes refers to the inverse limit in the "generical" way (referring to the definition) and sometimes it refers to the inverse limit as the above one.
Hence the conclusion I got is: I can think the inverse limit I prefer, depending on the particular situation I'm treating with. Am I right?
Thank you all.
These things are not on different sides of the coin. As you was already told, you met a concrete instance of the notion of categorical limit.
What is a categorical limit: in a few words, whenever you are given an inverse system and a family of compatible maps (you can package these two things asking the almighty gods for a functor $\varphi\colon (I,\le)\to \mathcal C$) you can find a "best approximation" for an object $L = \varprojlim X_i$ which is "near" to every $X_i$ in the sense that you have projections $\{L\to X_i\mid i\in I\}$ along which every other family $\{K\to X_i\mid i\in I\}$ factors uniquely.
This is a vertiginous generalization of the notion of product: whenever you are given a bunch of "things in a box" $\{X_i\}$ with no relation between them, you can form (provided your ambient is nice: f.d. vector spaces are nice as soon as $I$ is finite, but the product $\mathbb{K}\times \mathbb{K}\times \mathbb{K}\times \cdots$ doesn't exist) a product $\prod_{i\in I} X_i$; if there are relations between the various $X_i$, encoded in morphisms $X_i \leftrightarrows X_j \rightleftarrows X_k$ you have to take into account them: so you do not take all the elements in the product $\prod_{i\in I} X_i$, but only those elements which are compatible with the "compatible maps" $\varphi_{ij}$. That's the meaning of the explicit construction of $\varprojlim X_i$ as a subset of the product $\prod X_i$.
Sparse examples:
Before you ask, yes: you call them limits because they are unique, and before you ask, yes: $\lim$its are special kinds of $\varprojlim$its, but (IMO) it's boring and cumbersome to see why this is true. Oh, last but not least, before you ask: you write $\varprojlim$ because there's also a notion of $\varinjlim$: the latter are called colimits, and you define them taking quotients of coproducts $\coprod X_i$ (and initial objects, coproducts, coequalizers... are all examples of colimits).
I seriously hope you understood the nitty-gritty under the (invaluable) uniqueness property of limits: once you have an $L$ as above which satisifes the Universal Mapping Property (UMP), whatever it is $L$ it is isomorphic to $\varprojlim X_i$. More is true: $L\cong \varprojlim X_i$ in a unique way, i.e. with a unique isomorphism.
That's nice, since you have no choice at all but you're free as a bird: everybody can do his favourite Mathematics, and in the end not only our universal constructions are isomorphic, but there is a unique way to find these isomorphisms. Imagine two apparently different languages (Spanish and Russian?) where though every translation is bound to the same kind of uniqueness: UMPs tell you that to any mathematical purpose we are speaking the same language.
Mathematics is made (or discovered) in such a way that you don't have to worry about what $L$ you choose, as soon as it satisfies the right UMP, and that's a big, big luck!
Once you are familiar with this kind of constructions, you can package large areas of Mathematics under the same language: that's called Category Theory, and I strongly suspect you have been exposed to these things since AGLQ, but no-one told you the trick (golden rule for the math teacher...).
Ok, now for something more on-topic: if $I$ is a direct set, and you are working in a category of "finite widgets" (finite sets, finite groups, finite dimensional $K$-vector spaces, finite complexes of $R$-modules, topological spaces which admit a cellular decomposition in a finite number of cells, etc.) it is interesting for various reasons to study the "completion" with respect to directed limits of your category of finite widgets. When finite groups are completed you obtain precisely pro-finite groups, as you know. There's an interesting phenomenon going on here: pro-objects sometimes inherit a natural topology (that's the profinite topology).