Ok, this is a pretty long post that explains in a (hopefully) elementary way the intuition behind the notion. This is at least my understanding (I personally prefer filters over nets), if someone disagrees with it I welcome criticisms.
A net is just like a sequence, but instead of indexing with $\mathbb{N}$, you do so with a more general ordered set, that is supposed to be directed (often you see a definition of nets where the directed set is not supposed to be ordered, but I never saw the point, if someone knows what purposes it serves I would be interested).
The point is that in a topological space that doesn't have countable neighborhood bases, sequences are not enough to detect continuity of a function, so you want to find a finer notion which works in the same way as sequences, but capture more information on the topology of the space.
In order to define convergence, what you truly need is to be able to say "this [thing] is eventually in such and such neighborhood". The core of convergence - and asymptotic behaviour in general - of sequences is that something happens "if you go far enough in the indices".
So you want define a net on a topological space $X$ as a function $I\rightarrow X$ with $I$ replacing $\mathbb{N}$ and having enough structure to make sense of things happening eventually. So you need to be able to tell that one "moment" (index) is later than some other, that is you need to have a notion of comparison of $I$ : it is an ordered set (or preordered if you really want to).
Then you may want to imitate the notion of eventually that exists for sequences : something happens eventually for $(x_i)_{i\in I}$ if there is some index $k$ such that for all $i\geqslant k$, it happens for $x_i$. That works fine if $I$ is totally ordered ; but you need not impose that. If $I$ is not totally ordered, that definition has a flaw : suppose you have some index $j$ that cannot be compared to $k$, and furthermore cannot be compared to any $i\geqslant k$. Then your notion of eventually becomes a little awkward since it means that you can follow any chain of indices from $j$ as far as you want and never become greater than $k$, so your event that is supposed to happen eventually needs not happen for any index greater than $j$. This is embarrassing. Imagine $I$ is the union of two copies of $\mathbb{N}$, no element of one copy being comparable to any element of the other. Then the above solution of eventually will only restrain what happens in one of the two branches, the other being completely decorrelated. That won't do.
So you add the condition that for any two indices $i$ and $j$, there is an index $k$ that is further than both $i$ and $j$, ie $i\leqslant k$ and $j\leqslant k$. This prevents the weird things mentioned before from happening.
Now why is that a good enough notion ? The trouble with sequences was that to reflect faithfully the topology of $X$, you need the order structure on $I$ to be at least as fine (in some sense) as that of the neighborhood of any point. No $I$ will universally do the job : whatever $I$ you choose, you will be able to build topological spaces complicated enough for $I$ to be too coarse to capture its topology. But if you allow $I$ to be anything, then for any topological space you will get $I$ that are fine enough ; indeed, the neighborhoods of a point form a directed ordered set for (reversed) inclusion, which is tautologically fine enough.
This culminates in the theorem that a function is continuous iff $f(x_i)$ converges whenever $(x_i)$ converges (where converge means as usual "is eventually in any neighborhood of some given point"). And a whole bunch of other similar theorems.
Now a subnet is just like a subsequence : for a subsequence you restrict $(x_n)_{n\in \mathbb{N}}$ to a subset $J$ of $\mathbb{N}$ that contains numbers "far enough", in the sense that for any $n\in \mathbb{N}$, there is some $m\in J$ greater than $n$ ; this allows $J$ to retain some asymptotic information. Actually the formal definition is that you choose some increasing function $\varphi : \mathbb{N}\rightarrow \mathbb{N}$ such that its image has the above property, and define the corresponding subsequence of $(x_n)$ to be $(x_{\varphi(n)})$.
For a net, you put exactly the same definition with a directed set $J$ and a function $\varphi : J\rightarrow I$ (increasing and such that any element of $I$ is lower than some element of $\varphi(J)$). Note that it would have been natural to restrict ourselves to $J\subset I$ and $\varphi$ being some inclusion map, but it turns out not to be enough (but it requires pretty vicious counterexamples).
