In stockmarketese, a position is a holding in a stock.
By buying a stock you are taking a position in it. So any stock you own, you now have a position. "Joe," you tell your whip-smart friend at work, "I just took a position in Cisco."
"Really?" Joe inquires. "Because I sold my position at the height of the market in March 2000 -- I haven't taken any position in Cisco since."
The term is often used at Fool.com when an investor owns some stock in a company, but not all that she wishes to acquire -- a partial position. Many Fools like to buy partial positions with the goal of eventually building out their full position. This calls to mind the buying in thirds approach.
The full position amount is whatever you are comfortable investing in a stock. Remember to stay diversified -- most Motley Fool advisors advocate that investors hold 15 or more full positions.