When we're talking about autologin we're referring to the system that recovers and restarts a users session after she did a page refresh, or closed the browser and later reopened it and the OX frontend. If you want to find out more about the session recovery system, please refer to OXSessionAutologin.
The auth-id is a unique ID that allows tracking login/logout requests across the systems in your apache / OX cluster. It is useful for finding out which login request passed through which systems.
The cookie hash (not the other way around) is a unique string identifier computed from certain aspects of the login request. It is used to a) bind the session to certain client characteristics to prevent a session overtake and b) to allow session data of more than one session to be stored in the same cookie store (by providing unique names for the cookies). See also OXSessionLifecycle.
The client is a string identifier, used to identify a client that wants to use a session. Its values follow the reverse-domain-name convention of java packages, so for example the OX frontend identifies itself as com.openexchange.ox.gui.dhtml. The client is usually passed as a parameter to the login call and becomes one component of the cookie hash.
The form login describes a login call that can be triggered by an external and custom form. This was formerly (pre 6.20) known as the easy login. If you want to provide your own login form, this is the way to go about it. You can read all the details here.
The IP check describes a security check the OX server uses on sessions. Upon session creation the clients IP address is stored along with the session data. Later accesses within the session must then come from the same IP address that created the session. This is used to make session takeovers harder. See also OXSessionSecurityFeatures.
The session id is a unique string identifying the session. The session id, together with the session secret, is used to verify the authenticity of a session. In subsequent requests the session id will usually be transmitted as a request parameter, but might be stored in a cookie when the autologin feature is used.
The session secret, along with the session id, is used to verify the authenticity of a session. It is always passed to the OX server as a cookie. Only when the session id parameter and the session secret cookie belong to the same session will a request be accepted by the OX backend. The cookie name has the format open-xchange-secret-[hash] with 'hash' being the cookie hash. If you want to know more about the session secret and how it is used to verify the authenticity of a session, please refer to OXSessionLifecycle and OXSessionSecurityFeatures.