The paired ttest is used to compare the values of means from two related samples, for example in a 'before and after' scenario.
The difference between the means of the samples is unlikely to be equal to zero (due to sampling variation) and the hypothesis test is designed to answer the question "Is the observed difference sufficiently large enough to indicate that the alternative hypothesis is true?". The answer comes in the form of a probability  the pvalue.
Worked Example
Consider the following study in which standing and supine systolic blood pressures were compared. This study was performed on twelve subjects. Their blood pressures were measured in both positions. It is therefore, a paired samples design.

Subject

Standing

Supine

Difference

1

132

136

4

2

146

145

1

3

135

140

5

4

141

147

6

5

139

142

3

6

162

160

2

7

128

137

9

8

137

136

1

9

145

149

4

10

151

158

7

11

131

120

11

12

143

150

7

Mean

140.83

143.33

2.50

SD

9.49

10.83

5.50

The statistical analysis of paired data is performed on the differences between the pairs, and for this data the mean difference (Supine  Standing) between the blood pressures is 2.50 mmHg. The standard deviation (SD) of the difference is 5.50 mmHg.
Suggested null and alternative hypotheses could be:
H_{0}: There is no difference between the mean blood pressures in the two populations
H_{1}: There is a difference between the mean blood pressures in the two populations
or equivalently
H_{0}: On average there is no difference between the blood pressures in the two populations
H_{1}: On average there is a difference between the blood pressures in the two populations
The computer output from performing a paired samples ttest on the standing and supine blood pressure data gives a pvalue of 0.144. Thus the probability of getting a difference of 2.50 mmHg between the mean blood pressures (given that position does not affect blood pressure) is 0.144 or 14.4% or about 1 in 7. This is not sufficiently low to conclude that position does affect mean blood pressure. Therefore, we fail to reject the null hypothesis with this data, and conclude that there is insufficient evidence to suggest a difference between blood pressures, on average, in the two positions.
The paired ttest
The output for the standing and supine blood pressure example is shown below, and gives the sample means and the mean difference, their standard deviations and the standard errors. The 95% confidence interval for the mean difference is also shown as well as the ttest of the null hypothesis that the "mean difference = 0" versus (vs) the alternative hypothesis that the mean difference is "not = 0". The pvalue equals 0.144.
In SPSS
Enter the data from the first sample into one column and the corresponding data from the second sample in a second column, then select
Analyze > Compare Means > PairedSamples T Test...
The output is as follows:
in Minitab
Enter the data from the first sample into one column and the corresponding data from the second sample in a second column, then select
Stat > Basic Statistics > Paired t...
Paired TTest and Confidence Interval
Paired T for Supine  Standing

N

Mean

StDev

SE Mean

Supine

12

143.33

10.83

3.13

Standing

12

140.83

9.49

2.74

Difference

12

2.50

5.50

1.59

Assumptions underlying the paired sample ttest
Both the paired and independent sample ttests make assumptions about the data, although both tests are fairly robust against departures from these assumptions.
In the paired samples ttest it is assumed that the differences, calculated for each pair, have an approximately normal distribution. Techniques are available to test this assumption. An alternative procedure that makes no assumptions about the distribution of the data is the Wilcoxon Test, but this test is less powerful than the paired sample ttest.