Typical personal finance advice would have you divide your budget categories into two groups: Your fixed expenses and your discretionary expenses. I generally don't like that distinction much--how is your power bill more fixed than your grocery bill? When you reach the point of emergency economizing, though, it's a useful way to structure your thinking.
Fixed expenses can be reduced, but those reductions often require long lead times (waiting for a lease to run out, so you can move to a cheaper place) or they require an upfront investment (buying a more energy-efficient refrigerator). Even when you can cut them, it's generally not practical to reduce them to zero, (except for those few people living off-the-grid).
So, in a financial emergency, the first place to look is at your discretionary expenses.
Start with zero
If you've got a budget, go through the non-fixed expenses and plug in zero for every number. Then, go back and adjust up the ones that are really essential. (Starting from zero and just budgeting what your household needs is, by the way, always a good idea.)
You've got to have food. In an emergency, though, you can cut your food bill by a lot more than you probably think. (And, as a bonus, cheap eating is probably healthier than what you were eating before, even if it may have to be less organic and less local.)
You've got to stay healthy. If you've got medical insurance, keep it in effect if you possibly can. If you're being treated for a medical condition, call your doctor's office and inquire if the treatment you're getting is the lowest-cost treatment that's available. (The phone conversation--typically with a nurse or physician's assistant--will probably be free.) In a financial emergency, it probably makes sense to delay even things that are important, if they're not urgent--eye and dental exams, routine check-ups, etc.
If you've still got a job, you probably need transportation. In an emergency, though, you should zero out any transportation expense that isn't earning you money. Every trip should be either to work or from work, with stops for errands along the way and not trips of their own. Reducing the number of cars your household supports can save a huge amount of money--each car costs thousands of dollars a year in fuel, insurance, and financing expenses. Consider things like carpooling, public transport, walking, bicycling, and so on.
Education is tricky. If you're not in school, zeroing out your education budget doesn't save you much money. If you are in some sort of degree program, disrupting it might reduce your future earnings by vastly more than you're saving--and yet, that might be the right choice in an emergency. On the other hand, if financial aid is paying most of your education expenses, or if your health insurance depends on your being a full-time student, your education expense may be too good a bargain to pass up if you can possibly afford it.
Debt payments generally can't be escaped, except by filing bankruptcy, which is obviously a last resort. There may be a few exceptions--student loans can be deferred under certain circumstances, there are moves afoot to develop programs for restructuring mortgages. For debts that are tied to some specific thing (such as a car), consider selling the thing and putting the money toward the loan. Doing that will generally leave you out of pocket, but getting the monthly payment off your back can still leave you ahead--the sort of hard choice you sometimes have to make in an emergency.
That's about it. Every other discretionary expense should go to zero: recreation, eating out, vacation, travel, clothes, shoes, etc.
Defer what you can't avoid
Many expenses that can't be avoided can be deferred in an emergency. Generally, don't replace (or pay to repair) things that break or wear out. For example, instead of getting a broken dryer fixed, dry your clothes on a drying rack until the emergency is over. (As a bonus, the clothes will last longer.) You probably can't get away without fixing your furnace or hot water heater, but you can get by without lots of things that you're used to using every day--microwave, toaster, TV, stereo, iPod, etc. Make them last as long as you can, but when they go, do without until the emergency is over.
Sometimes proper maintenance will save a lot of money in the long run if done promptly--replacing a roof before there's water damage--but in an emergency, it's often necessary to accept that you won't be able to make the choice that's cheapest in the long run, because you're short of cash in the short run. That's the nature of emergencies--you do what has to be done, and then do what you can to mitigate the harm after the emergency is over.
Ask for necessities
If you have relatives who give you gifts, ask that they give you necessities instead of luxuries. Nobody wants to get socks and underwear for Christmas--except people with holes in their socks and worn-out elastic in their underwear.
Use your time
If your financial emergency is due to the loss of a job, you've now got time that you didn't used to have. Some of it--most of it--should probably go toward finding a new job. But there's still time that can be used in place of spending money. Cook cheap meals from scratch (much cheaper than prepared meals). Do stuff around the house that you might otherwise have hired someone else to do. See if you can't do stuff for neighbors--helping in their garden, showing them how to create a website, and tutoring their kids can keep you on a more even footing when they're sharing produce from their garden, giving you rides into town, and letting you use their tools. Make things (sweaters & scarves, jellies & jams, beer & wine, cakes & pies) that you can give as gifts or barter for stuff you need but can't make.
Don't dismiss the fixed expenses entirely. For one thing, even small measures like adjusting your thermostat and turning off lights you're not using will definitely save dollars, even though they won't reduce your utility bills to zero. For another, even "fixed" expenses are only fixed in the short term. Depending on how long your emergency lasts, some (or even many) of your fixed expenses will become unfixed. Know when your lease is up, when your cell phone contract is up, when the term ends for your kid's private school. Look especially at annual fees that will get charged automatically if you don't cancel something. Figure out now when you'll have to put the wheels in motion in order to switch to a lower-cost option at the next opportunity.
If you're not in debt, it's pretty amazing how low you can push your expenses on an emergency basis, simply by zeroing out all your discretionary spending and deferring other spending (including essential spending) until the emergency ends. As in so many other areas of life, it's really debt that's the killer. In fact, I'd go so far as to suggest that, in the current economic situation, having any significant debt means that you're already in a financial emergency--even if you've got a good job and a solid emergency fund. I recommend some preemptive belt-tightening and getting that debt paid off.