Let’s assume for a moment that we compared number of hours studied by a group of college students to their final exam scores. We might assume that those who studied longer hours would score higher on the final exam. That is considered a positive correlation.
Now, let’s think about hours exercised per week by a group of college students, and how that might relate to percent body fat. More hours exercised would reasonably lead to a lower percent body fat, right? (Actually, that’s more of a thesis question, but for the sake of my example let’s say yes.) That is considered a negative correlation.
Further reading here.
Yes, you really have to do all those tests.
Much of statistical analysis is based on the assumption that you have a lot of data to work with and the data follow a normal distribution. Beyond that, though, there are reasons we can’t just work up a t-test and a Pearson correlation without examining the data set first.
Non-parametric statistical tests are appropriate in different instances. This article from Boston U School of Public Health is a good place to start if you’re wondering.
In any instance, it’s always a good idea to run a normality test (Kolmogorov-Smirnov or Shapiro-Wilk, depending on sample size) before further analysis.