Competition experiments are designed to determine whether one genotype has a fitness advantage over another.
Although fitness can be measured in a number of ways, competitive fitness assays using pairs of bacterial strains are particularly simple if you can tell the two strains apart, either by some sort of marker (such as GFP) allowing you to count cells, or antibiotic resistance or colony morphological differences which allow you to differentiate cells on agar plates.
The underlying expectation is that the fitter strain will produce a greater number of viable off-spring than the other strain in a given period of time. The relative difference can be calculated a number of ways, but most simply the initial numbers and the final numbers are used (we use competitive fitness (W) = ln(Xf / Xi) / ln (Yf / Yf) where X and Y are the two strains, W is expressed as the competitive fitness of X compared to Y, f are the final numbers, and i the initial numbers). This method is essentially comparing the Mathusian ratios of the two strains (or genotypes).
There are some limitations or issues with this type of approach: both strains need to grow during the assay, and if one does not (or even dies off) you can get a meaningless negative result (you can redo the analysis to look at the selection coefficient instead to get around this); and you need a constantly growing population (you need to design an experiment which will allow good growth over the period of the assay); the competitive fitness is averaged over the assay period (it might change during the assay as the relative ratios of the two strains change); and genetic changes (mutations) occurring in one or the other strain may confound things (strains might diversify to produce more than the original two genotypes).
You could also determine fitness by comparing the growth rate of your two strains independently, but there is an added advantage to competitive fitness assays, because the two strains need to compete with one another for limited resources, deal with any toxic residues or antagonistic compounds which might be produced, as well as dealing with abiotic factors such as limited nutrients etc.
I am only starting to appreciate how powerful competitive fitness assays are in providing some meaningful ecological understanding of whether differences between two strains (e.g. different alleys of the same enzyme) makes any biological difference. We often spend a lot of time looking at sequence differences and postulating whether or not they make a difference, and competitive competition assays (undertaken in an ecologically-relevant environment) that can provide this sort of answer.