In automata theory, we all read automata as finite automata, from the very beginning. What I want to know is, why are automata finite? To be clear, what is it in an automaton that is finite - the alphabet, language, strings made with regular expressions, or what? And are there (in theory) any non-finite automata?
All automaton models you'll typically encounter are finitely represented; otherwise there would be uncountably many, which means they are not captured by Turing-complete models. Or, in CS-think, they'd be useless¹.
"Finite automata" are called finite because they only have a finite set of configurations (the input string aside). Pushdown automata, for instance, have a stack that can have arbitrary content -- there are infinitely many possible configurations.
Nota bene: Configurations of PDAs are still finitely represented! In fact, any computational model that falls inside Turing-computability has to have finitely representable configurations, otherwise TMs wouldn't be able to simulate them.
- I consciously disregard hypercomputation here for the purpose of this question.
The full name is "finite state automata". The crucial part is that the state of the automaton can be fully characterized by an element of some finite set of discrete states. This means that if the (relevant) state of the automaton involves a real-valued variable, there is an infinite number of potential states (putting aside the finiteness of floating point representations), and the automaton is not finite.
This might not ever happen in theoretical computer science, but it's not too exotic in the domain of modeling real number sequences. Hidden Markov Models can be used to model a number sequence as the output of a probabilistic system consisting of one output filter for each state of a (discrete, finite) Markov model with unobservable states. The filters compute the next real-valued output from a vector of the previous outputs ("autoregressive" model), but the underlying Markov model is finite since its transition probabilities only depend on the current Markov state.
However, this article (full text) develops a variation where the transition probabilities also depend on the real-number vector of prior outputs. The state of this system is the combination of a discrete Markov state and a tuple of real numbers ("mixed state Markov model"), hence it can be in an infinite number of different states.
In short, non-finite automata are theoretically well-defined, and occasionally even encountered.
In a finite automaton, there is quite a bit of finiteness: the number of states, the size of the underlying alphabet, and the lengths of the strings accepted by the machine.
You can certainly relax the finiteness condition on these by allowing, say, the machine to have infinitely many states but if you do, the resulting machine becomes uninteresting, in the sense that such machines can be designed to accept any languages at all.
In fact, in automata theory (which departs a lots from the origins of Kleene, Rabin and Scott), there are many forms of automata that are not finite. This arises for several reasons.
Pushdown automata, for instance, are automata that have an infinite set of configurations (these have a finite number of states, but the reality is that these should be thought as 'infinite automata').
In the same vein, there ar other examples of infinite automata for which the state space is infinite, but with a lot of structure. For instance one considers the class of automata that have as state space a (finite dimension) vector space, and as transition functions linear maps (plus some initial an final things). These are known as weighted automata over a base field (due to Schützenberger in 61). These can be minimized and tested for equality. Other examples include register automata (these automata have a finite set of registers, and work over an infinite alphabet: these can compare letters with registers and store letters in registers), and the more modern form of nominal automata (that have the same expressiveness, but have better foundations and properties). Emptiness of such automata is decidable.
But, even for studying finite state automata, it makes sense to talk about infinite automata. Indeed, consider the category of finite state deterministic automata that accept a fixed given language L, equipped with the standard notion of automata morphism, then this category fails to have an initial and a final object. However, if you live in the category of all deterministic (say countable) automata, then there is an initial object (the automaton that has as states $A^*$, as initial state the empty word, when it reads letter $a$ in state $u$ it goes to state $ua$, and a state is accepting if it belongs to L). There is also a final object (that has as states languages!). The existence of these two objects are one way to explain at high level why deterministic automata can be minimised and is tightly linked to the Myhill-Nerode congruence.
To conclude, there are infinite automata, but the models that are first studied in a lecture are always the finite state ones.
I think the question is assuming the conclusion that there are no infinite state automata when there are, they just aren't worth bringing up.
In Automata Theory, there is a hierarchy of powers of different virtual models. The one I learned had 4 (that I remember, it has been some time), the one I found on Wikipedia has 5. The weakest in either finite state automata, and the strongest is Turing machines. There is a some concept of a level more powerful than Turing machines that is loosely called hypercomputation. Many different descriptions of virtual machines fall into one of these levels. Turing machines are especially known for numerous models all having the same level of computational power.
An infinite state automata, at least one formally defined, will fall into one of these levels. There may something a little interesting in showing how a certain rigorous definition of an infinite state automata fits into one of these levels, but other than that, it wouldn't be of much use given there are far more well studied virtual machines representing each level. It's similar to how there isn't much interest in a billion tape Turing machine, since it would be no more powerful than a single tape Turing machine but more complicated to deal with.
Now, if you happened to find an infinite state automata that wasn't equivalent to an existing level of the hierarchy, that might be interesting. But without that, infinite state automatas are already captured within the existing hierarchy, and given the added complications of dealing with infinity, we just avoid them in the same way we avoid any overly complicated Turing machine model.
The (naive) infinite version of deterministic finite state machine is too powerful. Such a thing would be capable of "memorizing" any language at all, so there is not much to be learned from considering them.
For example, over a binary alphabet, consider an automaton in the form of a complete binary tree of infinite height. Every possible string you could consider as input corresponds uniquely to a node of the tree, so you can construct a decider for any language at all simply by making the corresponding nodes accepting states.
There is such a concept as an "infinite" automata, however, they are not normally as interesting as finite automata practically to software or hardware designers. There is some research and research papers on infinite automata, but I'll give a simple (and practical example) below, and you may see why no-one talks about infinite automata except for perhaps, mathematicians, but I dare say some of them have interesting properties.
Given some event you want to count, then the infinite automata yields the output, where for the sake of simplicity it's state directly maps to the traditional output symbol in an alphabet (giving the answer). Both state and output symbol space is infinite in this case, and is the set of natural numbers [0..infinity)
count = 0 count --- [event] ---> count + 1
Essentially this is "counting". It is however also a state machine, but it's so trivial that we say "counter".
(Note that computers may a have limit on the size of the counter, but the premise of the machine is still valid.)
Things may get more interesting with real values and complex mathematical functions, for most people, these are mathematical functions, and not state machines. The marking thing to note about a state machine, however, is that they can be in a given state, and change state based on some external input, like the counter above, and they do so over time through a sequence of inputs.
Infinite automata, that is, like finite automata but with an infinite number of states, are a particular kind of transition system. Infinite transition systems of various kinds are commonly used to describe the behaviour of discrete systems.