Federated peg is described in Appendix A of the sidechain whitepaper, https://blockstream.com/sidechains.pdf . I don't completely understand the details but here is an overview. In order to implement sidechains in the ideal fashion, the Bitcoin protocol must be modified. It is difficult to gain consensus for such a modification. However, in the meantime, "federated peg" is proposed as a temporary measure for implementing sidechains.
The idea of sidechains is to set up a system that allows you to effectively transfer Bitcoins from the main chain onto the sidechain, and then later transfer those Bitcoins back from the sidechain onto the main chain. While the Bitcoins are 'effectively' in the sidechain, from the perspective of the main chain they are just sitting in some address ('locked'). I like to imagine that the 'locked' Bitcoins on the main chain are 'sleeping' deep in a trance, projecting their 'astral body' into the 'astral plane' of the sidechain. When they finish their business on the sidechain they can then later 'return to their body' and 'wake up' on the main chain.
With federated peg, in order to transfer the Bitcoins from the main chain to the sidechain, you transfer it to an address on the main chain and give control of that address to a small set of trusted parties using a multisignature transaction (eg perhaps you choose 5 trusted parties and then use a 3 of 5 multisig; then in order to move the coins back to the main chain, 3 out of 5 of them will have to agree). These trusted third parties are called 'functionaries' in the paper.
Then, on the sidechain, the trusted third parties inform the sidechain that you have just transferred in those Bitcoins. The 'astral bodies' of your Bitcoins can then be transferred to others on the sidechain. Perhaps at some time in the future, a future owner of some or all of these 'astral bodies' wants to move them back to the main chain. This person informs the trusted third parties (the functionaries) of the main chain address that e wishes to send the coins to, and attaches a proof-of-ownership showing that they are the current owner of the 'astral bodies' on the sidechain. The functionaries inspect the proof-of-ownership and if it is valid, they then delete (or otherwise remove from circulation) the astral bodies from the sidechain, and use their multisignature authority on the 'locked' coins on the main chain to unlock them and send them to the requested address.
This method is called 'federated peg', presumably because it is a type of 'peg' (connection between coins on the main chain and coins on the sidechain) enforced by a 'federation' of trusted third parties (functionaries). This method requires trust in the trusted third parties (functionaries) because if 3 of 5 of them collude (or however many of them there are), or get their keys stolen, then the 'locked' Bitcoins can be stolen on the main chain. However, the goal is for this to just be a temporary measure, until such time as consensus is gained to modify the Bitcoin protocol to implement sidechains using a different method, called 'SPV peg'. After that, you would be able to stop using 'federated peg' and use 'SPV peg' instead; unlike federated peg, SPV peg allows you to transfer Bitcoins between chains without trusting anyone.
'Federated peg' does not modify the Bitcoin protocol or require the cooperation (or even knowledge) of anyone outside of the users of the sidechain, so it has no impact on the security of the main chain.