It is tough to be quick, but in general this term is used to describe an electronic device, aka a vaporizer that heats oils and e liquids. The entire unit is composed of a vape battery plus the atomizer or cartridge attachment. In this space you have rechargeable batteries that will last a long time but then there are also the ones that will be replaced every so often.
Technically, and rather loosely, the term can incorporate any vaporizer out there that accomodates oils and e liquids, such as a box mod with an oil atomizer attachment - that is not exactly pen shape or even super small, but it still falls under the umbrella of an oil vape pen in a loose way.
But, usually the oil vape a friend refers to comes equipped with a skinny 510 oil cartridge - these cartridges have become super popular thanks to the growth of the oil and dispensary industries. Take one oil cartridge and screw it into a vape battery and you now have a complete oil vape pen. These are the most popular thanks the the industry growth plus the convenience of using these cartridges - buy it, screw it on, and vape - no grinding and loading of herbs or sticky handling of wax. Most commonly these batteries are thin, cylindrical, stylus-style batteries that will last for a cartridge or two, maybe 3 and possibly more but expect the battery to go - however, the skinny 510 oil cartridge vape pen has evolved and cooler models are now available, although the best one is not exaclty pen shaped, but it is arguably the top vape battery for 510 oil cartridges - the SteamCloud Mini.
The terms oil and e liquid are used interchangeably and typically a vape for one will work for the other. So there really is no difference, just some specifics that separate the two. For instance, e liquid typically does not come in pre-filled cartridges, instead the e juice is poured from a bottle into a 510 threaded atomizer attachment - especially for big cloud blowing vapers, the quality atomizer attached to a box mod allows them to vape all day with the power to get huge hits.
On the other hand, other oils, not e juice, come in pre-filled cartridges and very rarely would a user manually fill a tank. These pre-filled cartridges are skinny, cylindrical in shape and work with a whole array of 510 batteries which includes box mods, evod style, micro and the usual skinny, stylus pen style.
The watts given off by a battery will directly impact the quality of the vapor. It will also affect the integrity of the atomizer or cartridge attachment. Too much power will lead to a hot, burnt tasting vapor that may leave you coughing and the throat feeling harsh. Also, too much power can burn coils out more quickly, and if you are dealing with a pre-filled cartridge it would be a disaster to burn out the coil before finishing. With a variable wattage battery a user can start with the power at the lowest setting and gradually increase the output as needed, but the session will always be adjusted to your preference - also some oils are thicker than others and therefore can take more heat, but again, the adjustable function will allow you to evauluate and adapt.
Often used with e juice but not always referred to as a pen, box mods are super popular and usually are of solid quality with the abiltiy to last a long time - both on a day to day basis as well as overtime. E liquid vapers are already aware of this but for those using the pre-filled cartridges, know that you have options, and a box mod is a great one that if used properly, will deliver the best experience around.