Let $g_1,..., g_t$ be real analytic functions defined on an open subset $U \subset \mathbb{R}^n$. Then $$ X = \{ x: g_1(x) = ... = g_t(x) = 0 \} $$ is called an analytic variety. In algebraic geometry, if all the $g$s where polynomials we expect the dimension to be $n - t$. Do we expect the same behaviour for analytic varieties?
Also what is a $0$ dimensional analytic variety? Is is a countable set? (in comparison to finite points in algebraic geometry?) Any comments are appreciated! Also I am looking for a reference for this. Thank you
Your statement about the expected dimension in algebraic geometry is problematic (really, you should say "for sufficiently nice collections of $g_i$," at least). Things get even worse once we move to analytic spaces: the function $\sum_{i=0}^r x_i^2$ cuts out an analytic space of dimension $n-r$ inside $\Bbb R^n$, so you really don't have much that's positive to say here about this question. In fact, any analytic variety can be written as the zero locus of a single equation: if we look at the variety cut out by $g_1=0,\cdots,g_r=0$, then the same variety is cut out by $\sum g_i^2$.
For the second question, if you work inside a copy of $\Bbb R^n$, then it is true that a zero-dimensional analytic variety must be at most countable. Any uncountable set in $\Bbb R^n$ has an accumulation point, so an analytic function vanishing on such a set is identically zero.
Before giving you a reference for more material on such things, let me make a brief discussion on the history of the field so you have some context. Frequently the objects you mention are also called real-analytic spaces ("variety" tends to come with a smoothness or reducedness assumption). They have not enjoyed the same popularity as some other closely connected fields - they were an active part of geometry in the 1950s and 60s (Hironaka proves resolution of singularities for analytic spaces as well as algebraic varieties, for instance) with Grauert, Remmert, Cartan, etc providing many contributions, but as Grothendieck-style algebraic geometry grew, they faded a bit from prominence. Adding to this the fact that real-analytic spaces were never as popular in the literature as complex-analytic spaces and there's a little bit of a dearth of writing out there about them.
From a perspective emphasizing the sheaf-related issues, one text I've found useful is Guaraldo, Macri, and Tancredi's Topics on Real Analytic Spaces (1986). One can also analyze such spaces via semi-analytic geometry and more generally o-minimal geometry, though one has to pay attention to the fact that analytic varieties don't actually form an o-minimal structure and must be enlarged a bit. I'm a little more familiar with this side of things, and one jumping off point I like is Denkowska and Denkowski's survey article A long and winding road to definable sets, perhaps followed by Bierstone and Millman's Semianalytic and subanalytic sets.