As Charco has said, the hydroxide is in equilibrium with the water through proton exchange.
The hydroxide does the attacking because it is the most nucleophilic (more nucleophilic than water certainly)
the reagents will be listed as NaOH/H2O because you must carry out the hydrolysis in water.
Using another solvent could cause issues with getting the right product.
Any suitable solvent would have to be able to stabilise OH- and Na+ ions so you'd have to use something polar, and ideally something that can hydrogen bond.
An obvious choice might be to use an alcohol solvent but this causes a couple of problems.
lets say the ester is ethyl ethanoate. If you chose methanol as the solvent you'd set up the equilibrium
CH3OH + OH- <---> CH3O- + H2O
and seeing as the methanol is in huge excess the equilibrium would be pulled all the way to the right, so you wouldn't get hydrolysis, you'd just end up turning your ethyl ethanoate into methyl ethanoate!
If you were to choose ethanol solvent then the same problem would arise, you get rapid exchange of the ethoxide group, but no hydrolysis (because the alcohol, in huge excess, would always substitute first).
If you chose a non alcoholic solvent, chances are, if it could dissolve NaOH and stabilise the resulting ions, it would probably be a nucleophile and would spoil your chances of hydrolysing anything.
So the NaOH increases the rate of hydrolysis because it is more nucleophilic, but the water as solvent is crucial in achieving the right product!
Sorry for the long explanation, hope that makes sense!