In political or policy network studies an actor is a primary variable used to measure power and influence based on the number and type of relationships they have with other actors.
The second primary variable is "ties" or "connections."
Often, actor traits (race, gender, ideology, nationality etc.) and structural traits (distribution of ties, tie strength etc.) are used to understand how the people involved in a particular political theatre influence others inside the network.
Methods in network analysis can be either qualitative (close examination of who knows who and why the relationships exist) or quantitative such as counts of ties (degree) or variations of these counts (brokerage, eigenvectors etc.).
Player gets used in game theory studies, to similar effect. Game theory studies typically explore various potential decisions of two or more players in a 'game' (for example, members of a cartel attempting to set a high oil price) and how much they will gain or lose given how each player acts.
In general, 'player' and 'actor' can be exchanged for each other. Network studies sometimes refer to players and game studies sometime refer to actors, although most political scientists like to keep terms consistent where ever possible. They basically mean the same thing, although player implies some form of competition is involved.