Potsherds, toolstone flakes or lithic debitage, stone tool and tool fragments, bone tool and tool fragments, burnt or charred bone, cordage - these are some of the types of artifacts you might see while exploring the Southwest.
An artifact is essentially any object made by or modified by human hand. This can include a simple river cobble picked up and used once as a hammer, a fragment from a broken pot, beads, or even the smallest flake created during the manufacture of a dart or arrow point.
When you are lucky enough to find an artifact or "relic", keeping in mind how fragile the item is, it is okay to pick it up, examine it, photograph it, make a drawing of it and then please put it back where you found it.
Again, common sense comes into play here. A whole or almost intact pot is going to be extremely delicate and should not be handled at all. The same would apply to any cordage, woven material, textile fragments or human remains. Blades, projectile points, manos or metates, hammer-stones; these are durable enough to touch and examine.
Obviously, if you happen to come across an artifact in the middle of a jeep road or heavily used trail, go ahead and move it a short distance off to the side, out of harm's way, but don't collect it! Otherwise, leave artifacts where you found them. If it's particularly unique, for example a complete stone tool or ceramic vessel, locate it as accurately on a map as you can and report it to the most convenient Federal or State land management office. There will likely be an archaeologist in the office that would be very grateful for the information.
Each individual artifact is part of a bigger puzzle that will tell the history of the area. Whenever someone collects one for their personal gain, that piece to the puzzle is lost. Imagine buying a new 1000 piece jigsaw puzzle and discovering that dozens of pieces were missing. You end up with an incomplete picture. The same applies to the archaeological record of any given area.