I use silk thread, which I buy in neutral colors - it blends in and is practically invisible. For needles, I use either sharps or straws, depending on how I feel.
In the past, I've used Coats and Clarks, cotton floss, and 60 weight cotton. If you match the color of the thread to the applique the stitches will still be hard to see, which is what you want...unless you're doing the button hole stitch. For that, I've used hand quilting thread and cotton floss. I think a lot of quilters use pearl cotton for the buttonhole stitch, too.
You can find silk thread online - I think Keepsake Quilting has it, among others. My local shop didn't carry it for a long time, and now they only carry a few colors. It is expensive, but a spool lasts a real long time.