I think its fair to start by saying the Saints Row IV is still a bit of a rip-off of Grand Theft Auto. You're a crime boss, you run around a big, open city picking up dynamic missions from the world, and can spend a session following up the plot or driving around with the radio on picking fights with the cops or climbing buildings or trying to hijack that perfect car. However what SRIV remembers, which GTA seems to have forgot, is that this is inherently very, very silly, and rolls with in a major way. Everything about the game is arrowed in to making it as joyously stupid as it can possibly be. In a good way.
So lets start with the plot. After a Call of Duty spoofing opening full of barked commands to open doors and kill "terrorists", it transpires that you are, in fact, now the President of the United States for some reason. Which is pretty cool, but then aliens invade, blow up the planet, and enslave you in computer simulation of the city from SRIII, because at least that way the developers can save some time. Due to your own outstanding awesomeness, however, you're able to free your friends, subvert the simulation, and start to develop superpowers.
I think that sums up the tone the game is going for.
There is a feeling of finale to the whole game - publisher THQ went bust just before this released, so there may never be another Saints Row game, and this one is littered with call backs to the earlier games in the series. No character, alive or dead, seems to be left un-revived, and whilst a lot of it was lost on me as a series newbie, it did at least allow for the game to indulge in some fun spoofery. You get to "romance" your team mates in between missions as per Mass Effect, there are Smash TV style killer gameshows, some Tron, a great bit of Street Fighter style 2D beat 'em up and even an odd, Dance Dance Revolution bit. It's all very strange.
It's also sadly a little unfinished. The game is buggy at times, with missions simply not completing or starting, and the networking was very flakey despite using Steam's usually excellent matchmaking. Quest tracking frequently went awry, and more than once I got clipped into a building I then couldn't get out of. SRIV aims to get around this by letting you get stupidly overpowered very quickly, so the fun factor outweighs the frustration, and the auto-saving at least, is efficient and meant I rarely lost any significant progress. But it did feel rushed out, which in many ways it was.
So in the end Saints Row IV manages to be exactly what it appears - dumb, derivative, and extremely good fun. It remembers, at every stage, that it wants to be escapist, and over-the-top, and makes sure that it is. It never bothers you with the terrible things you did in the Old Country, nor nag you going bowling. It doesn't want to tell you a story based on watching The Wire too many times. It just wants to entertain you, and let you blow things up dressed as a nun wielding a alien disintegration rifle, if that's the sort of thing you want to do.
Which I did. Boom.