Let's make a list here. Everyone is invited to add and complete the list and the proofs.
List
0) locally compact Hausdorff spaces $\longleftrightarrow$ commutative C*-algebras
0') proper continuous maps $\longleftrightarrow$ non-degenerate C*-homomorphisms $A\to B$
0'') Continuous maps $\longleftrightarrow$ non-generate C*-homomorphisms $A\to M(B)$
1) compact $\longleftrightarrow$ unital
1') $\sigma$-compact $\longleftrightarrow$ has a countable approx. unit ($\iff$ has a strictly positive element)
2) point $\longleftrightarrow$ maximal ideal
3) closed embedding $\longleftrightarrow$ closed ideal
4) surjection/injection $\longleftrightarrow$ injection/surjection
5) homeomorphism $\longleftrightarrow$ automorphism
6) clopen subset $\longleftrightarrow$ projection
7) totally disconnected $\longleftrightarrow$ AF-algebra (AF = approximately finite dimensional)
8) One-point compactification $\longleftrightarrow$ unitalization
9) Stone-Cech compactification $\longleftrightarrow$ multiplier algebra
10) Borel measure $\longleftrightarrow$ positive functional
11) probability measure $\longleftrightarrow$ state
12) disjoint union $\longleftrightarrow$ product
13) product $\longleftrightarrow$ completed tensor product
14) topological K-Theory $K^0$ $\longleftrightarrow$ algebraic K-theory $K_0$
15) second countable $\longleftrightarrow$ separable w.r.t. the C*-norm
Proofs
0),1),2),3),5) follow directly from Gelfand duality - details can be found, for example, in Murphey's book about C*-algebras. For 0'), see here (I wrote this up because I didn't know any reference). A C*-homomorphism $A \to B$ is nondegenerate if the ideal generated by the image is dense. For 4) see here. 6) is given by characteristic functions. A reference for 7) is Kenneth R. Davidson, C*-Algebras by Example, Theorem III.2.5. It is related to 6) because a commutative C*-algebra is AF iff it is separable and topologically generated by the projections. 8) follows from abstract nonsense and 1). 9) ?. 10) is the Riesz representation Theorem. 11) follows from 10). 12) asserts $C_0(X \coprod Y) = C_0(X) \times C_0(Y)$, which is trivial. 13) asserts that the canonical map $C_0(X) \hat{\otimes} C_0(Y) \to C_0(X \times Y)$ is an isomorphism - this follows from the Theorem of Stone-Weierstraß. 14) is the Theorem of Serre-Swan.
One of my favourite examples is the following theorem, due to S. Mori:
Theorem A. Let $X$ be a smooth complex projective variety such that $-K_X$ is ample. Then $X$ contains a rational curve. In fact, through any point $x \in X$ there is a rational curve $D$ such that $$ 0 < -(D \cdot K_X )\leq \dim X+1.$$
In other words, smooth Fano varieties over $\mathbb{C}$ are uniruled.
The proof of this beautiful result uses deformation theory in a very striking way. The idea is the following. One first take any map $f \colon C \to X$, where $C$ is a smooth curve with a marked point $0$ such that $f(0)=x$.
Now by deformation theory of maps one knows that, if one requires that the image of $0 \in C$ is fixed, the morphism $f$ has a deformation space of dimension at least
$$h^0(C, f^*T_X)-h^1(C, f^*T_X) - \dim X = -((f_*C) \cdot K_X)-g(C) \cdot \dim X.$$
So, whenever the quantity $-((f_*C) \cdot K_X)-g(C) \cdot \dim X$ is positive, there must be a non-trivial family of deformations of the map $f \colon C \to X$ keeping the image of $0$ fixed. Then, by another result of Mori known as bend and break, one is able to show that at some point the image curve splits in several components and that one of them is necessarily a rational curve passing through $x$.
Instead, when $-((f_*C) \cdot K_X)-g(C) \cdot \dim X$ is not positive we are in trouble. But here comes another brilliant idea of Mori: let's pass to positive characteristic! In fact, in positive characteristic we may compose $f \colon C \to X$ with (some power of) the Frobenius endomorphism $F_p \colon C \to C$. This increases the quantity $-((f_*C) \cdot K_X)$ without changing $g(C)$ and allows us to obtain a deformation space which has again strictly positive dimension. So, using the argument above (deformation theory of maps + bend and break), for any prime integer $p$ we are able to find a rational curve through $x_p \in X_p$, where $X_p$ is the reduction of $X$ modulo $p$ (for the sake of simplicity I'm assuming that $X$ is defined over the integers).
Finally, a straightforward argument using elimination theory shows that if $X_p$ admits a rational curve through $x_p$ for every prime $p$, then $X$ admits a rational curve through $x$, too.
It is worth remarking that no proof of Theorem $A$ avoiding the characteristic $p$ reduction is currently known.
This kind of argument was first used by Mori in order to prove the following theorem, which settles a conjecture due to Hartshorne:
Theorem B. If $X$ is a smooth complex projective variety of dimension $n$ with ample tangent bundle, then $X \cong \mathbb{P}^n.$
See [S. Mori, Projective manifolds with ample tangent bundle, Ann. of Math. 110 (1979)].
More details about Theorem $A$ (as well as its complete proof) can be found in the books [Debarre: Higher-dimensional algebraic geometry] and [Kollar-Mori, Birational geometry of algebraic varieties].
Best Answer
I like the book by Chriss and Ginzburg (Representation Theory and Complex Geometry, https://doi.org/10.1007/978-0-8176-4938-8) very much, and I think it fits many of your requirements.