Homodyne and heterodyne detection
I'll assume AM DSB modulation for simplicity, but the same idea applies to quadrature modulation. I'll assume that the received signal has been properly bandpass filtered. I'll also ignore noise and carrier synchronization.

So, the received signal is $$r(t) = s(t)\cos(2\pi f_c t).$$ We want to recover the baseband signal $s(t)$ from $r(t)$. The bandwidth of $s(t)$ is $B$ and the bandwidth of $r(t)$ is $W=2B$. The signal $r(t)$ is centered on $f_c$; we would like the receiver to accept $f_c$ as a parameter set by the user (we say that the receiver should be tunable).

A homodyne receiver calculates $$\hat{s}(t) = \text{LPF}\lbrace r(t) \cos(2\pi f_c t) \rbrace.$$ While mathematically this is super simple, in practice this receiver is extremely difficult to implement. This was especially true in the early age of radio, where the entire system was analog and the theory was not yet well understood.

The heterodyne receiver (invented by the unsung, tragic genius Armstrong, as pointed out by Stanley) recovers $s(t)$ in two (or more) stages. First it calculates
$$s_{IF}(t) = \text{BPF}\lbrace r(t) \cos(2\pi f_1 t) \rbrace,$$ where $f_1$ is selected so that $s_{IF}(t)$ is centered at frequency $f_{IF}$.  In a second stage, the receivers calculates $$\hat{s}(t) = \text{LPF}\lbrace s_{IF}(t) \cos(2\pi f_{LF} t) \rbrace.$$ The point of this receiver is that the second stage is independent of the chosen value of $f_c$. This single idea simplifies the implementation so much that it is actually the reason we've had radio at all for the past 70-80 years.

If you're interested in this subject, there are two books I can't recommend enough:


"The science of radio", by Paul Nahin.
"Man of high fidelity", a biography of Armstrong by Lawrence Lessig.


On the technical side, I like the presentation of different modern receiver architectures in


"RF system design of transceivers for wireless communications", by Qizheng Gu.