I got a HP Compaq 615 and I wanna improve it's wifi connection and I found this dongle on ebay http/
Is it faster than the one in my compaq?
I want it to be as small as possible and that dongle seems pretty small so yea...
Maybe. That USB adapter is a "N" adapter. If your router is an "N" or "AC" router then it might be faster. Your laptop is only "G" which is slower than "N".
BUT a tiny adapter like that has almost no antenna so you may not get the same range as you do with your current built-in adapter.
that will entirely depend on which wireless card your PC shipped with: http/support.hp.com/us-en/product/Compaq-600-Notebook-PC-series/3958411/model/3958412/document/c01768616#AbT4
click wireless devices and 3 cards that shipped with that unit are 54Mbps devices while one is a 300Mbps capable.
if you have one of the 54Mbps cards then yes it will be faster, theoretically, than the laptop. if you have the 300Mbps card then the dongle will be half as fast, theoretically. this is assuming that the infrastructure can operate at that speed.
ultimately you will be limited by the router/AP/device that you connect to.