How do Bitcoin clients find each other? What protocols or algorithms are Bitcoin clients using in order to find their peers? I learnt that BitTorrent clients may rely on central computers called trackers. Do Bitcoin clients need any kind of central servers or nodes in order to operate?
Bitcoin clients use several methods to locate other clients. The primary method is a list of nodes from a previous connection to the network. The works very well for everything but your first connection or a connection after a very long period of disconnection.

For the case where you have no previous known IPs or they aren't usable, the primary fallback method is by DNS seeds. Several host names (such as dnsseed.bluematt.me) are maintained that resolve to a list of IP addresses known to be running nodes. Should that fail, the client has a list of IP addresses believed to point to stable nodes hard-coded into it.

One method that used to be used, but that has since been abandoned, was having the clients locate each other using an IRC server. That worked similar to a BitTorrent tracker.

Once you are connected to the network, you will begin to receive lists of IP addresses and ports of known clients.