Gnosticism was one of the biggest and most widespread heresies in the early church. Because of this, it’s also a complicated one, with lots of different versions of Gnosticism, enormous volumes of writings and strange apocryphal gospels, and so it’s hard to give a fair account of its details. There are three main strands in what we count as Gnosticism, though, which go as follows:
Firstly, the idea that this imperfect world is separates from the Supreme Being by a series of intermediaries or ‘aeons’. There are different opinions on how many aeons there are, and what exactly their nature is, but they generally all have names. This idea they got partly from Plato and partly from other Greek philosophers including Pythagoras, who thought it was really important that God was One and simple. This then created the problem of how you get from the One and simple God to the multiple and complex world, and as the gnostics didn’t want the simplicity of God to be spoiled, they placed these intermediaries between God and the world.
Secondly, the Gnostics thought that our souls were perfect entities that had somehow got trapped in human bodies. When people realised that this was what had happened, they naturally wanted their souls to be set free, which they did by mastering each aeon dividing their souls from the Supreme Being, which meant that they needed to know the names of the aeons. The Gnostics claimed to know these secret names.
Thirdly, the Gnostics thought that it was possible for souls to escape and be set free, and that this was made possible by redeemers who appeared at different points in history to reveal the secret knowledge needed to escape the physical world. They saw Jesus as one of these redeemers.
Apart from the fact that it’s pretty weird (and quite like Scientology, I think, though definitely less expensive), there are several problems with Gnosticism from a Christian point of view. Firstly, they were really secretive with their ‘special knowledge’ - you had to get in the club to be told how to be saved. They were less like missionaries and more like the Masons, probably with the strange initiation rites to boot.
Secondly, they were often quite elitist. A lot of them thought that not everyone had a ‘divine spark’ for a soul, but it was only a few special people who could be saved. You had to be the ‘right sort’ of person to get in, unlike Jesus’ kingdom which welcomes the tramps, the prostitutes, and the tax collectors. I think it's fair to suggest that this is a tendency Christianity hasn't entirely lost (though, to be fair, we did let you in).
Thirdly, and this is the biggy, they thought that the world was something bad that kept us from God, that our bodies were evil, and that creation as a whole wasn’t made by God himself but by some lesser, probably evil being. It’s this form of Gnosticism that pops up most often throughout the history of the church, and which undermines both the fact that God created the world and declared it good and also the Incarnation, which suggests that salvation comes through and in the physical world, not despite it. You can see Gnostic tendencies today in the Church's sometimes funny attitude to sex but also, I think, in wider society's attitude to food (ooh, isn't this chocolate cake sinfully good. Oh, I'm so naughty).