A contract state (or just "state") contains opaque data used by a contract program. It can be thought of as a disk file that the program can use to persist data across transactions. States are immutable: once created they are never updated, instead, any changes must generate a new successor state. States can be updated (consumed) only once: the notary is responsible for ensuring there is no "double spending" by only signing a transaction if the input states are all free.
A participant is any party that is able to consume this state in a valid transaction.
The list of participants is required for certain types of transactions. For example, when changing the notary for this state, every participant has to be involved and approve the transaction so that they receive the updated state, and don't end up in a situation where they can no longer use a state they possess, since someone consumed that state during the notary change process.
The participants list should normally be derived from the contents of the state.