For some people, it should be a top priority. For others, there are higher priorities. What it should be for you depends on a number of things, including your overall financial situation (both your current finances and how stable you expect them to be over time), your level of financial "education", the costs of your mortgage, the alternative investments available to you, your investing goals, and your tolerance for risk.
Your #1 priority should be to ensure that your basic needs (including making the required monthly payment on your mortgage) are met, both now and in the near future, which includes paying off high-interest (i.e. credit card) debt and building up an emergency fund in a savings or money-market account or some other low-risk and liquid account. If you haven't done those things, do not pass Go, do not collect $200, and do not consider making advance payments on your mortgage. Mason Wheeler's statements that the bank can't take your house if you've paid it off are correct, but it's going to be a long time till you get there and they can take it if you're partway to paying it off early and then something bad happens to you and you start missing payments. (If you're not underwater, you should be able to get some of your money back by selling - possibly at a loss - before it gets to the point of foreclosure, but you'll still have to move, which can be costly and unappealing.) So make sure you've got what you need to handle your basic needs even if you hit a rough patch, and make sure you're not financing the paying off of your house by taking a loan from Visa at 27% annually.
Once you've gotten through all of those more-important things, you finally get to decide what else to invest your extra money in. Different investments will provide different rewards, both financial and emotional (and Mason Wheeler has clearly demonstrated that he gets a strong emotional payoff from not having a mortgage, which may or may not be how you feel about it). On the financial side of any potential investment, you'll want to consider things like the expected rate of return, the risk it carries (both on its own and whether it balances out or unbalances the overall risk profile of all your investments in total), its expected costs (including its - and your - tax rate and any preferred tax treatment), and any other potential factors (such as an employer match on 401(k) contributions, which are basically free money to you). Then you weigh the pros and cons (financial and emotional) of each option against your imperfect forecast of what the future holds, take your best guess, and then keep adjusting as you go through life and things change.
But I want to come back to one of the factors I mentioned in the first paragraph. Which options you should even be considering is in part influenced by the degree to which you understand your finances and the wide variety of options available to you as well as all the subtleties of how different things can make them more or less advantageous than one another. The fact that you're posting this question here indicates that you're still early in the process of learning those things, and although it's great that you're educating yourself on them (and keep doing it!), it means that you're probably not ready to worry about some of the things other posters have talked about, such as Cost of Capital and ROI. So keep reading blog posts and articles online (there's no shortage of them), and keep developing your understanding of the options available to you and their pros and cons, and wait to tackle the full suite of investment options till you fully understand them.
However, there's still the question of what to do between now and then. Paying the mortgage down isn't an unreasonable thing for you to do for now, since it's a guaranteed rate of return that also provides some degree of emotional payoff. But I'd say the higher priority should be getting money into a tax-advantaged retirement account (a 401(k)/403(b)/IRA), because the tax-advantaged growth of those accounts makes their long-term return far greater than whatever you're paying on your mortgage, and they provide more benefit (tax-advantaged growth) the earlier you invest in them, so doing that now instead of paying off the house quicker is probably going to be better for you financially, even if it doesn't provide the emotional payoff. If your employer will match your contributions into that account, then it's a no-brainer, but it's probably still a better idea than the mortgage unless the emotional payoff is very very important to you or unless you're nearing retirement age (so the tax-free growth period is small). If you're not sure what to invest in, just choose something that's broad-market and low-cost (total-market index funds are a great choice), and you can diversify into other things as you gain more savvy as an investor; what matters more is that you start investing in something now, not exactly what it is.
Disclaimer: I'm not a personal advisor, and this does not constitute investing advice. Understand your choices and make your own decisions.