Before you start running your Tor relay, you need to know which different operation modes there are and which one you want to use.
All Tor relays transport packets within the Tor network. This is how it guarantees anonymity. In addition to that, you may configure your relay to provide additional services.
An exit-node relay (or just exit-node) is a relay that sends packets to the servers other clients of the Tor networks want to connect to. This makes you the Exit Point for the network. In other words, your router's external IP is the one the traffic will appear to come from. It will decrypt the received packet from the Tor network and the packet will get on the Internet.
Because the servers will see the IP address of your Tor relay, this may be cause trouble, depending on your region.
This is the most simple operating modes and very safe. Your relay will just pass packets around within the Tor network and will never be visible for hosts outside the Tor network.
This new operating mode is for people who want to connect to the Tor network from a country where Tor is censored. Governments use public lists of Tor relays and block them, so that nobody from within the country may connect to one of the relay. If you cannot do that, you cannot use Tor.
Bridge nodes are simple relays, but they are not in the public list of relays, so that they are usually not blocked. Tor clients may connect to bridges which will then forward their packets to the relays, which enables those clients to use the Tor network.
The Private Bridge mode is essentially the same as the Bridge mode, but the bridge won't be advertised so, that you can pass the IP address of your private bridge to people who want to use it, but it will not be used by the public.