The mixed-model design ANOVA gets its name because there are two types of variables involved, that is at least one between-subjects variable and at least one within-subjects variable.
The mixed-design ANOVA model (also known as Split-plot ANOVA (SPANOVA)) tests for mean differences between two or more independent groups whilst subjecting participants to repeated measures. Thus, there is at least one between-subjects variable and at least one within-subjects variable.
For example, are there differences amongst the heights of males and females at age 10 and age 20 years?
- Gender (male or female) is the between-subjects variable
- Age (10 or 20 years) is the within-subjects variable
- Of interest are the main effects for Gender and Age, and the Gender-Age interaction effect.
- This could be described as a 2 x (2) mixed-design ANOVA
Assumption testing 
- One or more within-subject variables e.g., time (pre-exercise and post-exercise pulse rates)
- One or more between-subject variables e.g., gender
- Sample size - ideally, at least 20 cases per cell
- Normality - Distribution of the DV (e.g., pulse rate) for each cell is normal
- Independence: Each participants' responses are sampled independently from each other participants' responses (e.g., this can be satisfied by random selection).
- Homogeneity of variance: Cells have similar variances.
- Sphericity: Population variances of the repeated measurements are equal and the population correlations among all pairs of measures are equal. Tested by Mauchly's. Violation increases Type I error rate. If violated, interpret adjusted results (e.g., Greenhouse-Geisser).
- Homogeneity of inter-correlations: Tested by Box's M: "The assumption ... is that the vector of the dependent variables follow a multivariate normal distribution, and the variance-covariance matrices are equal across the cells formed by the between-subjects effects." (SPSS 14 Help - Tutorial)
- See also these lecture slides
See also 
- Mixed between-within subjects ANOVA (allnurses.com)