List of genes across species (or just for humans)
In principle there are a LOT of these databases and depending on your specific questions you may want to look into different ones. Like already mentioned NCBI hosts basically all of the available information, however it's not always easy to find the bits you care about.

The database I personally use the most is biomart (hosted by the ensembl consortium). It doesn't just contain all known for genes for pretty much all organisms, but is also able to link this to a ot of different information sources.
I usually access the database directly via the biomaRt R package, so I can't really say how good the actual website this. However, this link should allow you to download a table of all human genes with names, (unique) identifies &amp; descriptions (I selected these from the 'Attributes' section).