What is a SAML profile, and which profiles exist besides Web Browser SSO?

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Accepted Answer

A profile bundles assertions, protocols and bindings into a precise recipe for one use case; Web Browser SSO is the most widely used, others include Enhanced Client or Proxy (ECP), Single Logout, Identity Provider Discovery, Artifact Resolution and Assertion Query/Request.
The building blocks alone leave too many choices for two vendors to interoperate. A profile fixes them: "use the Authentication Request protocol; send the request by Redirect, POST or Artifact; deliver the response by POST or Artifact; the assertion must contain a bearer subject confirmation with these fields; the SP must check these conditions".
Notable profiles:

Web Browser SSO: login through an ordinary browser.
Enhanced Client or Proxy (ECP): SSO for smart clients that can talk to the IdP directly (desktop apps, proxies), using the PAOS binding.
Identity Provider Discovery: how an SP finds out which IdP a user belongs to (the "where are you from?" page).
Single Logout: global logout across all session participants.
Assertion Query/Request and Artifact Resolution: server-to-server retrieval.
Name Identifier Management and Mapping: managing pseudonyms.

On top of the OASIS profiles, federations publish deployment profiles (for example the Kantara saml2int profile) that narrow the options further so any two members interoperate out of the box.
Go deeper:

SAML 2.0: Profiles (Wikipedia) — the main profiles with flow descriptions.
 OASIS: SAML V2.0 Profiles (PDF) — every OASIS profile normatively.
 Kantara: SAML V2.0 Interoperability Deployment Profile — the federation-level profile that pins down remaining options.