Alcohols have labile protons which are exchangeable through acid-base equilibria with other similar protons, such as other alcohols, amines, amides, thiols and especially water. This reaction is fast enough, on the NMR timescale, that it can be observed as a reversible exchange process. The most common scenario is that there will be some amount of water in the sample, and so the alcohol and water 1H atoms exchange. The reaction is typically so fast that you don't see the water and alcohol as separate peaks, but as a population-weighted average. Therefore the actual position of the peak depends on a number of variables; temperature, solvent, starting chemical shifts of the exchanging species, populations of the exchanging species, rate of exchange, solvent and so on.
When you say 'thicker', I assume you mean a broader linewidth. Again, this is a consequence of the exchange process, and that you are seeing an average peak for the -OH and (say) H2O. Sometimes this broadening is so large that you can only observe it as a broad hump in the baseline of the spectrum. Again, the linewidth will be a function of the starting chemical shifts of the two components, the rates of exchange, and their relative populations.
It is actually entirely possible to see the alcohol peak with a narrow linewidth, similar to the CH2 and CH3 linewidths. To do this, you need to be able to eliminate the exchange process, either by excluding water from the sample( run ethanol in very dry dmso for instance), or cooling down the sample sufficiently to slow down the exchange (probably not feasible for most common solvents with ethanol and water though). In this case, you will observe your alcohol as a narrow triplet (approx 1.7Hz coupling by memory), with the CH2 appearing as a doublet of quartets. A great example of observing alcohol protons is to run sucrose dissolved in pyridine-d5. All OH protons, and their couplings are observable.
It is really important to understand that the integrals of these peaks are still valid. In the absence of any exchange (where your alcohol peaks are sharp), the integration of an ethanol 1H spectrum should still give you 3:2:1. Where there is an exchange process occurring with (say) water, the integrals will be 3:2:X, where X=(sum of -OH plus water). This method of analysis can be used to verify alcohol content of spirits and wine, for instance.