Front-running is knowing an order is going to be inserted into an exchange's book, and sending your order before it; an example would be if a broker saw a client making a big trade and sent his order before the client's, hoping to profit when the market moved.

In most modern stock markets there is not tiered access. My order is just as good as anyone else's, and no participants are given unfair notice of another's orders. Firms can spend money and be more competitive in these markets (e.g. better analysts, faster systems, co-location), but anyone has a fair shot at competing. If an order is public, there's nothing wrong with me taking action based on it before you if my only advantage is superior technology.