Both panel data and mixed effect model data deal with double indexed random variables $y_{ij}$. First index is for group, the second is for individuals within the group. For the panel data the second index is usually time, and it is assumed that we observe individuals over time. When time is second index for mixed effect model the models are called longitudinal models. The mixed effect model is best understood in terms of 2 level regressions. (For ease of exposition assume only one explanatory variable)
First level regression is the following
$$y_{ij}=\alpha_i+x_{ij}\beta_i+\varepsilon_{ij}.$$
This is simply explained as individual regression for each group. The second level regression tries to explain variation in regression coefficients:
$$\alpha_i=\gamma_0+z_{i1}\gamma_1+u_i$$
$$\beta_i=\delta_0+z_{i2}\delta_1+v_i$$
When you substitute the second equation to the first one you get
$$y_{ij}=\gamma_0+z_{i1}\gamma_1+x_{ij}\delta_0+x_{ij}z_{i2}\delta_1+u_i+x_{ij}v_i+\varepsilon_{ij}$$
The fixed effects are what is fixed, this means $\gamma_0,\gamma_1,\delta_0,\delta_1$. The random effects are $u_i$ and $v_i$.
Now for panel data the terminology changes, but you still can find common points. The panel data random effects models is the same as mixed effects model with
$$\alpha_i=\gamma_0+u_i$$
$$\beta_i=\delta_0$$
with model becomming
$$y_{it}=\gamma_0+x_{it}\delta_0+u_i+\varepsilon_{it},$$
where $u_i$ are random effects.
The most important difference between mixed effects model and panel data models is the treatment of regressors $x_{ij}$. For mixed effects models they are non-random variables, whereas for panel data models it is always assumed that they are random. This becomes important when stating what is fixed effects model for panel data.
For mixed effect model it is assumed that random effects $u_i$ and $v_i$ are independent of $\varepsilon_{ij}$ and also from $x_{ij}$ and $z_i$, which is always true when $x_{ij}$ and $z_i$ are fixed. If we allow for stochastic $x_{ij}$ this becomes important. So the random effects model for panel data assumes that $x_{it}$ is not correlated with $u_i$. But the fixed effect model which has the same form
$$y_{it}=\gamma_0+x_{it}\delta_0+u_i+\varepsilon_{it},$$
allows correlation of $x_{it}$ and $u_i$. The emphasis then is solely for consistently estimating $\delta_0$. This is done by subtracting the individual means:
$$y_{it}-\bar{y}_{i.}=(x_{it}-\bar{x}_{i.})\delta_0+\varepsilon_{it}-\bar{\varepsilon}_{i.},$$
and using simple OLS on resulting regression problem. Algebraically this coincides with least square dummy variable regression problem, where we assume that $u_i$ are fixed parameters. Hence the name fixed effects model.
There is a lot of history behind fixed effects and random effects terminology in panel data econometrics, which I omitted. In my personal opinion these models are best explained in Wooldridge's "Econometric analysis of cross section and panel data". As far as I know there is no such history in mixed effects model, but on the other hand I come from econometrics background, so I might be mistaken.
