Relations between differential geometry and algebraic geometry I am currently an undergrad student looking to study some algebraic geometry, I have heard that differential geometry is useful for intuition in algebraic geometry, but I have no background in that (DG).
What are some specific examples of concepts that are common in both and would help understand the other subject? I am aware "manifold -> variety", but what else?
 A: I can give one important example: what a map is. You may recall that a map between smooth manifolds $f: X \longrightarrow Y$ is smooth if it is smooth under all charts. More formally, if we have $\phi: U \longrightarrow \mathbb R^n$ and $\psi: V \longrightarrow \mathbb R^m$ in the atlases for $X$ and $Y$ respectively and such that $f[U] \subseteq V$, then we say $f$ is smooth relative to these coordinates patches if $\psi \circ f \circ \phi^{-1}: \mathbb R^n \longrightarrow \mathbb R^m$ is smooth. If this holds for all patches in the maximal atlases of $X$ and $Y$ then we say $f$ is smooth.
As you say, we want to translate some of our notions from manifolds to varieties. The definition I gave above for smoothness doesn't translate super well. I'll give you another equivalent definition that may start to ring some bells if you're familiar with algebraic geometry.
Let $C_X$ denote the "sheaf" of continuous functions from $X$ to $\mathbb R$. In other words, for all $U \subseteq X$ open, $C_X(U)$ is the set of all continuous functions $U \longrightarrow \mathbb R$. Now, given a continuous map $f: X \longrightarrow Y$, we get a "map of sheaves" $C_Y \longrightarrow f_* C_X$. Don't let this notation scare you - all I'm saying is that $f$ induces a map $C_Y(U) \longrightarrow C_X(f^{-1}[U])$ for all open subsets $U \subseteq Y$. This takes $g \mapsto g \circ f$. Now, we let $C_X^{\infty}$ be the sheaf of smooth functions to $\mathbb R$. Again, I just mean that $C_X^\infty(U)$ is the set of all smooth functions $U \longrightarrow \mathbb R$ for $U \subseteq X$ open. Smooth functions are continuous, so $C_X^\infty(U) \subseteq C_X(U)$. Of course the same holds for $X$.
So in that same sense, if we have $f: X \longrightarrow Y$ smooth and $g: U \longrightarrow \mathbb R$ smooth for $U \subseteq Y$ open. Then $g \circ f: f^{-1}[U] \longrightarrow \mathbb R$ is smooth. Using that same fancy notation, we get a map $C_Y^\infty \longrightarrow f_* C_X^\infty$. In summary, if a continuous map $f$ is in fact smooth then the map $C_Y \longrightarrow f_* C_X$ descends to a map $C_Y^\infty \longrightarrow f_* C_X^\infty$.
In fact, this last condition completely characterizes smoothness! Putting all this fancy notation away, all I'm saying is that $f: X \longrightarrow Y$ is smooth iff it is continuous and sends smooth maps on open subsets of $Y$ to smooth maps on open subsets of $X$. More formally, if for all $g: U \longrightarrow \mathbb R$ smooth, $U \subseteq Y$ open we have $g \circ f: f^{-1}[U] \longrightarrow \mathbb R$ smooth, then $f$ is smooth.
Now this is something we can work with. I won't do much in the way of rigor here; you can read something like Hartshorne for that. Take a variety $X$ over an algebraically closed field $k$. We have the intuitive notion that a regular map $X \longrightarrow k$ should look like a polynomial, and a regular map on an open subset $U \longrightarrow k$ should look like a rational function whose denominator is nowhere zero on $U$. What then is a map between varieties, say $f: X \longrightarrow Y$? Appealing to the above, it's something that takes regular maps on $Y$ to regular maps on $X$. More formally, it takes $\mathcal O_Y \longrightarrow f_* \mathcal O_X$, where $\mathcal O_X$ is the sheaf of regular functions. Put aside what exactly a regular function is. The point is that they're continuous functions defined in an algebraic fashion. To map from one variety to another and preserve this structure, we follow the analogy of smooth maps I gave above. This is how Hartshorne defines a map of varieties, modulo details.
We can take this a bit further by realizing that the set of continuous, smooth, or regular functions on an open subset actually forms a ring, and the map $g \mapsto g\circ f$ is a ring homomorphism. This will eventually lead to the definition of a map of locally ringed spaces, and hence of schemes, but that's too much for just this one answer.
