If you are sending it to a printer, pdf is more than likely the way to go. next would probably by tiff then png. These two formats are raster formats so they would print as the resolution you created the file at. Not all printer support ai but ai and pdf would be vector formats that print at the highest resolution of the printer regardless of the documents resolution.
Jpg even though it compresses to a nice small file, is a lossy format, in that data is thrown out to save space. I does not support transparency or layers.
png supports transparency but does not support layers. (Layers for these two formats are photoshop layers)
tiff and psd supports layers and transparency as well as editable text.
eps is hardly used any more but some older software requires it.
pdf supports transparency, editable text, layers (both illustrator and photoshop) as well as a host of other options like color profiles, animations, etc.
A pdf file can also embed the ai or photoshop file depending on which app created the pdf. Making it editable in that app.
In the end, you use what format(s) that the printer supports, which is why it is always a good idea to ask the printing shop.
If it is a printer in your home or office, then just print it instead of exporting it. (unless you need the file in another app like Indesign or photoshop)