Standard horse fence around here (Loudoun County, Virginia) has 3 or 4 horizontal boards with the posts set on 8' centers. I think the standard height is 4'. If I think of it this evening, I will put a tape measure on our own fencing.

Standard fence boards are1x6 rough-sawn oak boards. You want to stagger the joints, so you need 16' boards. The saw mill usually cuts them a few inches longer than 16' so you can cut each board to the exact right length. Definitely check that before you dig your holes! Nothing worse than having to re-dig the holes.

Use good quality, 16-penny galvinized nails with either a twist or a ring shank. The nails at Home Depot are crappy, IMHO. Look for a hardware store that sells Maze brand nails. Or find someone knowledgable about nails who can recommend a good brand.