How are dependent and independent samples different?

Dependent samples are paired measurements for one set of items. Independent samples are measurements made on two different sets of items.

If the values in one sample affect the values in the other sample, then the samples are dependent.
If the values in one sample reveal no information about those of the other sample, then the samples are independent.

Example of collecting dependent samples and independent samples

Sample the blood pressures of the same people before and after they receive a dose. The two samples are dependent because they are taken from the same people. The people with the highest blood pressure in the first sample will likely have the highest blood pressure in the second sample.
Give one group of people an active drug and give a different group of people an inactive placebo, then compare the blood pressures between the groups. These two samples would likely be independent because the measurements are from different people. Knowing something about the distribution of values in the first sample doesn't inform you about the distribution of values in the second.