The independent t-test is of the most well-known statistical tests (see Student's t-test). It is useful for looking at differences in the average between two groups when the variable being averaged is numerical and inadequately normal - for example, looking at the difference in height between countries. If you are looking at paired data e.g. pre and post-diet weight, you need to use the paired t-test.
If you want to look for differences between multiple groups, the ANOVA (Analysis of Variance) is required. If your data is not normally distributed, N is less than 20, or there are outliers, you should probably use the Mann-Whitney U.
The p value tells you how likely the difference observed would occur if drawing from the same population (i.e. where the groups didn't really have a relationship with the value being averaged). A small p value tells you that it would be rare to observe such a difference if the group doesn't have a relationship with the value being averaged. From this we might reject the null hypothesis in favour of the alternative hypothesis - namely, that there is a difference according to the grouping variable.
In the example below (based on false data for illustration only), we shouldn't be surprised the p value is very low. It is possible to see a large difference in average age and N is reasonably large for each group. In this case we could reject the hypothesis that nation has no relationship with age.
Be aware that there is a certain sensitivity about terminology around this area. According to a widespread convention, we shouldn't conclude that there is a relationship, only that we reject the null hypothesis (see Hypothesis testing). We might go so far as to reject the null hypothesis in favour of the alternative hypothesis. See Statistical hypothesis testing.
Another video is also available showing how to do an independent t-test using SOFA Statistics: https://www.youtube.com/watch?v=-mc_pLdd6Jg&list=UUFRr0ugWcqCfhLwJ5qBlzpQ