What is a “free parameter” in a computational model?

A free parameter is one that can be adjusted to make the model fit the data. If I make a model that says $A$ is proportional to $B$, there is one free parameter, the proportionality constant. If my model has a specific value of the proportionality constant, there are no free parameters.If I say that $A$ is a quadratic function of $B$, there are three free parameters, $a,b,c$ in $A=aB^2+bB+c$. That makes it easier to fit the data, even if my model is not correct, so it is less impressive.

The following marvellous quote from Freeman Dyson's "A meeting with Enrico Fermi" contains both an example for your first question and an answer to your second.

[Enrico Fermi] delivered his verdict in a quiet, even voice. . . . "To reach your calculated results, you had to introduce arbitrary cut-off procedures that are not based either on solid physics or on solid mathematics."

In desperation I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, "How many arbitrary parameters did you use for your calculations?" I thought for a moment about our cut-off procedures and said, "Four." He said, "I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk." With that, the conversation was over. . . .