How can I adjust a door latch so that it engages securely?
You should check the hinges, and make sure the screws are tight and that the hinges are not worn out. Evidence of a worn out hinge will be that the door hangs at an angle, which is likely to be visible along the top of the shut door. If the door appears to be crooked, or if it wobbles when you lift it, then you should replace the hinges before moving making further adjustments.

In order to move the strikeplate so that the door shuts, you may need to enlarge the hole. Sometimes the adjustment is minor and you might have trouble putting the screws close to, or half-way overlapping previous screw holes. If you have a wooden door jamb, you can plug the old strikeplate's screw holes with a dowel rod. You should make sure the dowel is tight fitting (maybe requiring a light tapping with a hammer); use a drop of wood glue with the dowel so that the repaired screw holes are nice and solid. Let the glue dry before piloting new screw holes.

Here's a how to video about adjusting a door so that it closes.

Edit- based on pictures above here are some more details:


Double check the hinges. If they are okay, then you need to move the strike plate about 1/8" toward 5:00.

Shopping list:


Sharp 1/2" chisel
Box cutter
Flat-head screwdriver
Hammer
1" paddle bit
1/8" dowel
Wood glue
Drill
1/8" drill bit
Dremel tool with cut-off disk