Paul has heard from Epaphras of beliefs the Colossians held which got in the way of their relationship with God. We can't know very much about them beyond what Paul says about them because there isn't anything else recorded about them. At most, we can speculate.
Paul starts his letter praising the Colossians for their faith and love. Colossians 1:7-8 (ESV):
... just as you learned it from Epaphras our beloved fellow servant. He is a faithful minister of Christ on your behalf and has made known to us your love in the Spirit.
But he quickly moves to acknowledging that they are immature in terms of knowledge—especially when it comes to praxis. Colossians (1:9-10 ESV):
And so, from the day we heard, we have not ceased to pray for you, asking that you may be filled with the knowledge of his will in all spiritual wisdom and understanding, so as to walk in a manner worthy of the Lord, fully pleasing to him, bearing fruit in every good work and increasing in the knowledge of God.
What follows (Colossians 1:15-23) is perhaps the most laudatory description of a person ever recorded. Jesus is placed front and center before the reader as the very key to coming before God holy and blameless. Then Paul shifts to his own work, which is to make the mystery of Christ known to the Gentiles via (in part) his writings. He defines the mystery several ways including: "the knowledge of God's mystery, which is Christ, in whom are hidden all the treasures of wisdom and knowledge." (Colossians 2:2-3 ESV) For Paul, it seems clear, any practice that is not informed by an understanding of Christ is, at the very least, suspect.
Here's a list of things Paul warns against in the order he references them:
- plausible arguments (2:4)
- philosophy (2:8)
- empty deceit
- human tradition
- elemental spirits of the world
- questions of food and drink (2:16)
- new moon [celebrations]
- Sabbath [observance]
- asceticism (2:18)
- worship of angels
- sensuous minds
- elemental spirits (2:20)
- human precepts and teachings (2:22)
And that doesn't include the more dense list in 3:5-8. Nor did I include the warning against those who would compel circumcision in 2:11-14. If the Colossians knew of Paul's letter to the Galatians or had heard the teaching it contains, they would also be aware of Paul's conflict with those who insisted on Jewish traditions.
Looking at the list of things that Paul warns about, you start to get the impression he's using a shotgun approach—blasting away at anything that might be causing problems in the Colossian community. We can categorize some of these as Jewish traditions and some as Greek philosophy. Greek polytheism was likely present as well.
A slightly more speculative suggestion is that Paul is warning against Pythagorean dynamism. What strikes me is the the phrase translated as "the elemental spirits" comes from the Greek word stoicheion <4747>, which is from the same root as the title of Euclid's Elements. Pythagoras established an esoteric philosophy, which was informed by his mathematical work, that may have been the basis for some 2nd century Gnosticism. He has been credited with a wide range of beliefs including mysticism, metempsychosis, religious observances, asceticism and so on. So it could very well be that Paul has heard that the Colossians have become involved with Pythagorean cosmology.
In any case, Paul stands against all sorts of praxis of the old self and encourages unity in Christ. Colossians 3:9-11 (ESV):
Do not lie to one another, seeing that you have put off the old self with its practices and have put on the new self, which is being renewed in knowledge after the image of its creator. Here there is not Greek and Jew, circumcised and uncircumcised, barbarian, Scythian, slave, free; but Christ is all, and in all.