Depth of field is determined mainly by your aperture: the smaller the f/number (more open the aperture), the smaller your depth of field. So, for example, at f/2.8 you'll have much less depth of field than you will at f/8.
So, metering does sort-of affect depth of field, because the meter chooses your exposure. If you're in a mode where the camera gets to choose the aperture for you (i.e. Shutter Priority, Program, or any full-automatic mode), then the camera will choose an aperture and your depth of field will change appropriately.
This is why I very frequently shoot in Aperture Priority mode -- depth of field is often (not always) something I want to control, so I set the aperture and let the camera choose the shutter speed and ISO for me.
However, you can use any of those three types of metering with any of your exposure modes (Tv, Av, P, etc.). I would generally stick with evaluative unless you have a reason to need to change modes (for example, center-weighted and partial are good when parts of your image are much brighter or darker than other parts, and you only want the camera to properly expose just one part).
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