Since Rebirth, things have been pretty shaken up for Superman. You see, the version of the character that has been starring in the latest run is from the old DC universe, before the Flashpoint event occurred.
Flashpoint was a reality altering event caused by The Flash, which rebooted everything. At the beginning of the new comic universe, which was known as the New 52, writers got to create a new interpretation of Superman, as well as loads of other heroes. This newer version was less of a Boy Scout and a bit more edgy. However, through several bits of comic book nonsense, (which took place in the world-smashing comic event, Convergence) the original Superman ended up in this world with the original Lois Lane. This was explained in Superman: Lois and Clark, an 8-issue miniseries.
They arrived to find that this universe already has a Superman and a Justice League and so, since he wasn't needed, the original Superman and Lois Lane settled down and began their family. They set up a farm out in the countryside and lay low, keeping to themselves. They also chose the new surname of "White", after all there can't be two Clark Kents and Lois Lanes who look identical!
Unfortunately, just before Rebirth the new Superman dies, after getting kryptonite poisoning. This occurred in "Final Days of Superman" because he absorbed energy from the fire pits of Apokolips and exposed himself to massive amounts of Kryptonite.
And so now, the original pre 52 Superman is once again the one and only Superman in the DC universe! However, he wouldn't be the only Clark Kent for very long... but we'll get to that!
The main difference is that this Superman is a father. Jonathan Kent Jr. (named after Superman's earth dad) is a great addition to the DC Universe in my opinion and I'm very glad that the book gives him such a spotlight. He has now assumed the title of Superboy and he is firmly established to be a huge part of Superman's life. He has also just begun co-leading his own book with Robin in "Super Sons"!
So Jon is doing well and even Lois has managed to get her job at the Daily Planet back! Superman is doing what Superman does best and everybody is pretty happy!
If you were worried about missing the New 52 Superman, then don't worry- because thanks to even MORE comic book nonsense, the two versions have merged.
You see, when original Superman revealed himself to the world, New 52 Superman died- but not long after that, Clark Kent showed up... and no one knew what was going on. Clark Kent is Superman but Superman had just died... so who's this guy? Well, after several issues of mystery in Action Comics, it is finally revealed to be Mr. Mxyzptlk an inter-dimensional magical imp.
So Mxyzptlk is being pretty nasty and kidnaps Superboy, holding him ransom to toy with Superman. You see Mxyzptlk had been captured by someone very mysterious and very powerful... and Superman did not come to his aid!
When Mxyzptlk escaped he had to disguise himself in order to hide from his captor. He used his abilities to convince himself that he was just regular Clark Kent... so once again, there were two versions of Clark. Superman used this to his advantage, helping to dispel the idea that Clark Kent is Superman. So the secret identity of Superman is once again in tact.
When Mxyzptlk regains his memories, he holds a grudge. There's a big showdown between the Superfamily and Mxyzptlk, where the Pre 52 Lois and Clark and the energy of the New 52 Lois and Clark were merged.
Now both character's histories have been combined. A "single reality, built from two". Basically, fans can pick and choose their favourite stories from both pasts and say yes, that happened to Superman!
But now, enough about what's happened, let's look to the future. The current run has had a lot of crazy stories so far and if they keep up this quality of story-telling, this could be one of the greatest runs on Superman ever!