I recently installed Xfce with Xubuntu (hardy) and I am new at this so I am wondering if I run Gnome applications such as Gedit, does that mean that Gnome is running as well as Xfce?
Nah, shouldn't start any GNOME stuff unless you use something like nautilus (nautilus --no-desktop should avoid taking over the desktop), if in doubt check over the process list. (Open a Terminal and poke over "ps ax")
Soulslack is right in the sense that some gnome applications (or the way the packages have been built) pull in all kind of dependencies. They will not necessarily slow down XFce, but lighter/nimbler apps do exist, so you will need to weigh whether the extras are worth the hassle. Gedit has some requirements, but does not require nautilus. This statement is for Mandriva, which shows that the application doesn;t require Nautilus. Other distributions may have created other dependencies. In my experience (Xfce for many years with many gnome apps) there are not many aps that require nautilus the application; there are numerous that require libnautilus, but that does not conflict with xfdesktop or thunar.
As to Gedit: There is a somewhat less functional but considerably lighter alternative: mousepad. Frankly speaking I wouldn't know what I miss in mousepad ~12 month after switching. This shows various light alternatives to typical gnome applications. Deciding whether to go pure Xfce or mixed Gnome is taste, so a personal matter -- I use mousepad instead of gedit, but still use eog instead of ristretto and brasero instead of xfburn.