What does Shin Megami Tensei (Games like Shin Megami Tensei IV) have to do with the Persona series (Persona 3, IV, etc)? How are they related, if at all?
The Persona series is a spin-off series of the main (Shin) Megami Tensei series of games. While the two series have no direct connection in terms of story, they do share a number of common thematic and mechanical elements, such as the recruitment, summoning, and fusion of demons. In the Persona series, of course, they're usually called "personas" instead of "demons" and are explained somewhat differently in the story. Many of the individual demons/personas themselves are present across both series as well.
While the main-line Shin Megami Tensei series generally features a heavier emphasis on combat and mechanics, the Persona series has generally been positioned more as a traditional JRPG series, with more focus on story and character interaction. The third and fourth Persona games also introduce social sim elements, which are entirely absent from the main SMT games.
Aside from Persona, there are other SMT spin-off series as well, such as Devil Summoner, Digital Devil Saga, Devil Survivor, and so on.
The main Shin Megami Tensei games have no story link to the Persona games. What they do have are mechanical links.
For example, where you can collect the persona cards in P3 and P4, you convince demons to join your party in the main SMT games. From there, though, you can fuse demons just like in P3 and P4, complete with skill inheritance and the like.
The Persona games are technically related to the mainline series, but they are really meant to be enjoyed as standalone games. The Persona games are a spin off of Shin Megami Tensei If, and Shin MegamI Tensei If is a spin off of the orginal series where the nuclear apocalypse that takes place in Shin Megami Tensei II never happens. In Shin Megami Tensei III Nocturne it is revealed that there are multiple universes that are connected. More information can be found here
P.S - I'm sorry if I got some of the details wrong, I haven't played/finished all of the games.