Elasticity of demand influences what goods and services the government chooses to tax and what price producers set.
On a graph, i) inelasticity is depicted by a steeper demand curve, and ii) taxation is always depicted as a shift to the left of the supply curve (note this is not always a perfect shift). If you sketch it, you'll see that consumers bear most of the tax when the goods are inelastic and producers bear most of the tax when the goods are elastic.
They will tax inelastic goods and services such as petrol, alcohol, and tobacco because they will not lose many consumers - effectively maximising tax revenue. If they taxed elastic goods, there will be a reduction in supply (due to increased cost) and, amongst other problems, is not as effective for maximising tax revenue.
You mentioned price ceilings and floors, so here's where they come in -
When a good is inelastic, a rise in price will lead to a decrease in quantity demanded, but will still net more revenue. Producers can take advantage of this and raise prices to increase producer surplus at the expense of consumer surplus. However, some inelastic goods that are essential for survival (e.g. food) may have price ceilings imposed by authorities to keep them affordable, so producers can't set the price ludicrously high.
Hope this was of some help. Excuse any typos, I'm braindead from cramming for exams.
