fixed effects vs random effects vs random intercept model

I read somewhere that a random intercept model is a type of random effect model.

Yes, this is correct. Random intercepts are random effects.

But I thought that in a fixed effect model we were also assuming random intercepts, one per each unit of interest.

No, in a fixed effects model, the fixed effects are, fixed. They don't vary by subject or any other unit of interest. The model will estimate one parameter for each fixed effect.

Also because we are looking "within" the unit, as in, at each unit. Can you please clarify?

I'm not sure what you mean here. With mixed models, the fixed effects estimates are "within". If you want to account for "between" effects then you can fit contextual effects by using group means, and offsets from group means.

Would you be able to clarify what is a mixed model as well please?

A mixed model is a model that has fixed effects, and random effects. For example, suppose we have repeated measures within subjects, and we have 6 subjects. We might fit the mixed effects model:

 y ~ X + (1| subject)

and this will fit a model with a fixed effect for X and a random effect (random intercept in this case) for subject.

We could also fit a fixed effects model:

y ~ X + subject

and this will fit a model with a fixed effect for X and fixed effects for each leel of subject.

Both models should provide a good estimate for X but as the number of subjects increases there will be better statistical power, and interpretabilty, with the mixed model.