It depends a bit on how the original design was done.
If it's embroidered thread, then embroidery is the best way to recreate it. Fabric markers can be a good way to draw on detailed, delicate patterns.
Fabric patches can be mimicked with iron-on transfers. Print the logo on the transfer paper and put it either directly on the clothing or a small bit of fabric which you sew on. You can also find a lot of accurate patches (for military or space costumes) online.
Larger screen-printed designs (t-shirt logos or similar) can be done carefully with paint or markers. Trace the design onto the fabric, then go over it with paint or marker. Use acrylic craft paints, a small brush loaded with only a little paint, and a lot of patience. Markers are another option; either fabric markers or regular art markers. Either way, be sure to heat-set the ink with an iron before washing.
You can applique a design on, which means sewing fabric shapes down on your base garment. I'd only do that if the original seems to be separate layers of fabric, because getting nice crisp edges is hard.
If you've got a specific project you're working on, post a reference photo and we can give more concrete suggestions.