To define the singular homology of topological spaces we define a sequence of topological spaces $\Delta^n$ and maps $r_i:\Delta^n \rightarrow \Delta^{n+1}$. I'm kinda skipping a lot of details because someone who is going to be able to answer my question has very likely already seen singular homology.
In algebraic geometry consider the varieties over the field $k$, $k\Delta^n=\{(x_0,...,x_n) \in \mathbb A^{n+1}_k: \sum x_i = 1\}$ and maps $r_i : k\Delta^n \rightarrow k\Delta^{n+1}:(x_0,...,x_n) \mapsto (x_0,...,x_{i-1},0,x_i ,...,x_n)$ , defined for $0 \leq i \leq n+1$.
Then for any variety $X$ over $k$ define $C_i(X)$ to be the free abelian group generated by $\text{Hom}(k\Delta^i,X)$ and the differential $C_i(X) \rightarrow C_{i-1}(X)$ is defined just like in singular homology, namely a map $\phi : k\Delta ^i \rightarrow X$ gets taken to $\sum \phi \circ r_i$ and we extend linearly to formal sums.
Then let $H^{\text{var}}_i(X)$ be the homology of $C_*(X)$ at $i$.
How do I compute the homology of some varieties? For finite fields $k$ I should be able to do this numerically... I think? But for fun cases when $k = \mathbb R$ I have zero clue where to even start, for example where do I start with something like $H^{\text{var}}_1(\mathbb A_\mathbb R^2 - \{0,0\})$? I know it isn't zero but that's about it.
I can't use machinery already existing for singular homology since many theorems are not obvious to prove in algebraic goemetry due to the rigid nature of morphisms of varieties compared to continuous functions.
I've managed to prove that $H^{\text{var}}_i (-)$ is homotopy invariant, where a "homotopy" is just a map $\mathbb A^1 \times X \rightarrow Y$ but thats about all I have.