Coefficient of Determination (R2)

$$R^2=1-\frac{S{S}_{res}}{S{S}_{total}}$$

Basically $R^2$ is the amount of variance explained away by the model, normalized by the total variance. 0.8 implies 80% of the total variance is explained away by the model.

Adjusted $R^2$ penalizes the use of multiple parameters to explain the model. It does this by multiplying a bias term to the unexplained variance, increasing it if a lot of parameters are used.

$$adjusted{R}^2=1-(1-R^2)\frac{n-1}{n-p-1}$$

Basically the unexplained variance $(1-R^2)$ is multiplied with relative degrees of freedom, a value greater than 1 dependent on the size of p, reducing the overall adjusted $R^2$ value.