So I hope I do not make myself look like a idiot but I will try to help explain this from my understanding of it. I should note, this is not my full time job so please be kind if I have something really off:)
These protocols, and I believe eap is a protocol, peap is protected eap, so it is probably the same protocol with additional encryption, are used to authenticate at layer 2. This means the client has to communicate authentication information to devices it is directly connected to. In the old days this was pretty basic, you dial in over a phone line and send your user name and password usually in clear text.
As Wireless came out they had the same problem to solve, they had to authenticate people to give access to the network so they used what they knew, ppp or other protocols that authenticate over layer 2, it has been too long since I used dialup so I do not remember the different options. The problem now was that it was a shared medium, unlike a phone line, so they had to develop better ways to protect the username and password, they started this in dialup as well but it was not a huge concern because most people did not worry about their line being tapped.
So they started to build these safer methods of authenticating over Layer 2, hence was born eap. Now of eap, I believe, eap TTLS is the most secure but the hardest to get configured right. EAP-FAST is a spinoff of TLS in that it also uses tokens/certs but instead of an out of band way to deploy them is negotiates this inband which may make it less secure, LEAP I believe is not recommended to be used and was mostly used for AP to AP communication. Anyway, all these use different methods of establishing a EAP session in a secure manner in order to transfer the username and password in an encrypted fashion. With the exception of TLS, TTLS and PEAP, the methods used establish keys through negotiation not through something defined out of band, of the out of band ones PEAP can be configured to not verify certs so that is why I believe it is less secure then TLS and TTLS. Once the encrypted tunnel, or keys, are established the username and password are sent using the inner protocol which can be clear text, mschap or some others, do not know them off the top of my head, the important thing for this discussion is they are encrypted at layer 2 to the access device, then encrypted via PSK from the access device to the RADIUS server. If the PSK is learned and the inner method is cleartext then anyone can sniff the wire and read the password in wireshark.
So the other side of this is centralized management, since these are layer 2 communications device cannot get to your backend RADIUS server directly to properly authenticate, so part of the role of 802.1x is that your client talks to the access device, wired or wireless, the access device then talks at layer 3(IP) to the RADIUS server that does the authentication and returns attributes for access control as well as authentication status, but it is the access device that sets the access control and negotiates with the client directly.
So how does this fit into today's security?
It allows secure centralized management
It offers encryption from end to end
It facilitates the secure granting of access control to networks, via vlan assignments and downloadable ACL's
It abstracts the authentication source from the device, that is the RADIUS server is never reachable from the device prior to authentication.
If done right it offers higher layer 2 security then wired connections (although this is debatable in my mind)
Lastly, the bolded question:
Is EAP a way for wireless protocols to agree on algorithms, and each device provides their own implementation conforming to this framework?
EAP is used to define keys to be used by your symmetric encryption algorithms, I believe as part of the EAP negotiation it also negotiates the symmetrical encryption algorithm as well. Of course this would happen after the EAP authenticates the user.
So it does 2 things, exchange username/password and securely transfer symmetric keys from negotiated algorithms.