Correct me if I'm wrong here:
Conceptually, there are four possible effects: Fixed intercept, fixed coefficient, random intercept, random coefficient. Most regression models are 'random effects', so they have random intercepts and random coefficients. The term 'random effect' came into use in contrast to 'fixed effect'.
'Fixed effect' is when a variable effects some of the sample, but not all. The simplest version of a fixed effect model (conceptually) would be a dummy variable, for a fixed effect with a binary value. These models have a single random intercept, fixed effect coefficients, and random variable coefficients.
The next tier of complication (conceptually) is when the fixed effect is not binary, but nominal, with many values. In this case, what is generated is a model with many intercepts (one for each of the nominal values). This is where you get the classic 'multiple lines' of a panel data model, where each of the 'options' of a fixed effect variable gets its own effect. The virtue of throwing all the different factor-specific data series into a single regression (rather than doing each factor of the fixed effect as its own regression) is that you get to pool the variance of all the different effects in one equation, and so get better (more certain) values for all of your coefficients.
'Tier three' of complication would be when the 'fixed effect' is itself a random variable, except that its effects are 'fixed' to affect only a sub-set of the sample. At which point the model would have a random intercept, multiple fixed intercepts, and multiple random variables. I think this is what is known as a 'mixed effects' model?
'Mixed effect' models get used for multi-level modeling (MLM), as the 'fixed effects' can be used for nesting one subset of data within another. This grouping can have multiple tiers, with students nested within classrooms, nested within schools. The school is a fixed effect on the classrooms, and the classrooms on the students. (The school may or may not be a fixed effect on the student, depending on the experimental design--not sure)
Panel data models are 'mixed effect' models, but use two dimensions for grouping, typically time and some sort of category.