At the risk of being pedantic and nit-picky (you did ask a question about terminology after all), I propose that you call these objects "interactive programs" rather than algorithms or computers.
Is a game not an algorithm
The challenger and adversary can certainly be described in terms of some programmatic/algorithmic logic.
But I think of an algorithm as a non-interactive thing that takes an input, gives an output, and then is done.
A cryptographic security game is more interactive than that. There are often several distinct rounds where information is exchanged, and the challenger usually provides a collection of subroutines/oracles that the adversary can call, many times. I think it's important to highlight this interactive nature.
Technically you could identify an interactive program with its "next action function," which is indeed a simple non-interactive algorithm.
For example, the challenger is characterized by an algorithm that computes the logic: "if the adversary is making kind of query, then respond with action."
In fact this might be how you would choose to define what an interactive program is, in terms of non-interactive algorithms.
Apart from such very formal definitions, it is rare to really think & talk about challengers and adversaries in this way, but occasionally it is useful to think of protocol specifications in this way.
Are adversaries and challengers computers
I think that a computer is a device that can do what a program tells it to do. Adversaries and challengers are computers that run specific programs -- the challenger for a particular security game runs a fixed program but we consider an adversary running an arbitrary program.
What is $n$ supposed to be a parameter of?
Being a parameter means: it's a value that is chosen externally, and is provided publicly to both the challenger and adversary. The security parameter is given as input to all algorithms in the cryptographic scheme (although this input is often not written explicitly). Usually the security parameter specifies the size of the keys, etc.
The security parameter is like a knob that you can turn. A higher value makes it a little more expensive to run the cryptographic algorithms, but a lot more expensive to break it. Turn the knob until you are comfortable with how hard it is to break the scheme, and that's the value that you use when running this scheme.