The z-test is based on the Normal distribution, while the t-test is based on Student's t distribution. Although the formulas for both are very complex, the essential difference is that while both give bell-shaped curves, the t-distribution has somewhat fatter tails. In addition, the t-distribution varies depending on the number of subjects, but the z-distribution does not.
Both tests assume that the data are independent; that means that the score that one subject gets has no influence on the score that any other subject gets. The exception is the paired t-test, which explicitly allows for paired data. For example, if you got the weight of 100 men at random, and 100 women at random, the data would be independent. But if you got the weights of 100 couples, the data would not be independent and the paired t-test should be used.
The z-test assumes that the sample size is "large." Studies have shown that, in most cases, having 30 or more subjects is enough to qualify as large. The t-test makes no assumption about sample size; indeed, the t-test accounts for sample size by having a parameter for degrees of freedom, which is sample size - 1.
The z-test assumes that the two samples have the same variance. The standard version of the t-test also assumes this, but there is a variation of the t-test designed for unequal variances.