As described by Wikipedia, BLS uses Diffie-Hellman in some way. I understand how Diffie-Hellman works in both its normal and elliptic curve forms. But what is the "pairing function"?
BLS signatures work in any so called gap group, i.e., a group where the computational version of the Diffie-Hellman (DH) problem - the CDH - is hard, but the decisional version of the DH problem - the DDH - is easy. Below I'm using the notation from the wikipedia article on BLS.
Just recall, that the DDH in a group $(G, g, r)$ (where $g$ is a generator and the group $G$ is of prime order $r$) is that, when given $(g, g^a, g^b, g^c)$, it's hard to decide whether $c=ab$.
If you look at the BLS signature verification, one checks whether the signature $\sigma$ for message $m$ has the form $\sigma=H(m)^x$ and writing it as $(g,h=g^x, H(m), \sigma)$, one sees that verification is to check for a valid DDH tuple (CDH needs to stay hard for unforgeability). Note that $H$ is a hash function modeled as a random oracle that maps strings to elements of $G$. So $H(m)$ can be written as $g^\gamma$ for some unknown $\gamma$.
Now, the existence of a pairing $e: G\times G \rightarrow G_T$ (symmetric for simplicity) exactly makes $G$ such a gap group. The check $e(\sigma, g) = e(H(m), h)$ in the signature verification exactly allows you to test if $(g,h=g^x, H(m), \sigma)$ is a valid DDH tuple. Using bilinearity of $e$, this should be easy to see, and I leave it as an exercise to you.
Pairings used in cryptography have evolved into a large (rather quickly changing) field and requires some time to study them. But the above symmetric pairing can be instantiated e.g., by setting $G$ as a subgroup of a supersingluar elliptic curve group, $e$ being the Weil pairing and $G_T$ a subgroup of the multiplicative group of a quadratic extension field of the curve's base field (you may want to look here for details). The choice of $H$ also depends on the setting, but it is crucial that $H$ is constructed such that one does not learn the discrete logarithm to the base $g$ from $H(m)$. You could look at the original BLS paper,who instantiate the scheme in a so called Type-2 pairing setting (in their setting, $G_2$ is the gap DH group). You may also want to look at this paper for how you can instantiate BLS in other Types of pairings.