Needing help in order to infer the statistical hypothesis tests performed in an old paper
Someone else can probably elaborate on how "peripheral lymphocyte culture" works, but this is what the statistical results suggest to me:


  The results suggested that the SCE in occupational workers was significantly higher than that in controls (11.31 vs 6.37, P &lt; 0.001).


This is a two-sample t-test. Occupational workers were pooled into one group and the mean value of SCE for those workers (11.31) was compared to controls (6.37). The null hypothesis is that the means are not different. They reject the null hypothesis. It would be nice if they included the standard errors for the means, so that you would have a sense for the amount of variation within the groups.


  The SCE in workers exposed to CTP and to COV was higher than that of control (10.27 and 12.58 vs 6.37) respectively.


This is likely an ANOVA with three groups (CTP, COV, and control). This test is kind of redundant with the first test, since the appropriate post-hoc test will tell you that CTP and COV are both significantly higher than control. But since they report only one P-value, then this is probably the overall F-test for the ANOVA. So all they can say with this test is "at least one group is different." You don't know if, for example, CTP and COV are different from one another. It's not clear from the text that they did a posthoc test (Tukey's HSD, for example), but I doubt it.


  In workers exposed to CTP and COV, there were no differences of SCE for smokers and nonsmokers (P > 0.05).


Considering only the occupational groups, the sample is divided into smokers and non-smokers. There was no significant difference in mean SCE between the groups. This is a two-sample t-test like the first one. The null hypothesis is that the means are not different. They fail to reject the null hypothesis.

It's also possible (but impossible to determine from the abstract alone) that they did a single, larger multiple regression. Properly coded, they would be able to, at once, test for occupational vs. control, CTP vs. COV vs. control, and smoker vs. non-smoker. It would be pretty dicey with such a small sample, so they probably didn't take that approach.

Why do we perform t-tests?

The null hypothesis of a two-sample t-test is that the means of two groups are not different from one another.

How do we know that our data follows a normal distribution? Shouldn't we perform tests in order to decide whether our data follows normal distribution and is of equal variance?

Assumptions of t-tests include normal distributions within groups and equal variance between groups. These should be checked prior to carrying out the test. We can assume that the authors did these tests, but they are very rarely reported.

In case the criteria needed to perform a t-test are not fulfilled should we choose a non parametric equivalent?

Non-parametric alternatives should be considered when the assumptions are not met. That being said, t-tests are pretty robust to violations of these assumptions.

Also the anova mentioned in the answer above is an N-way anova?

ANOVA in general is a test of equality among N groups. So you could think of a t-test as just a special kind of ANOVA on two groups (indeed, they are numerically equal).