Money.  Pure and simple.  In CIV III, you can expand (to the most part) as fast as you want and generally it pays off well to do so.  CIV IV, you'll go broke expanding too quickly and if you hit 0% science and are still in the negative, you'll start losing troops, workers, etc.

In my humble opinion, as you start to master economy in CIV IV, you will start mastering all the other aspects as they fall into place and your scores will rise.

Here are my suggestions on less-common, but important ways to make money and keep economy high.

1. Hamlets.  Obvious one, but coming from CIV III, I didn't like building them as I wanted to go crush fools with my production units.  Build one or two hamlets on average per city and maybe tell that city (or all cities) to emphasize economy in its governor if you find yourself needing money fast.  Biggest advice... build them next to streams as they already start with 1 gold per turn there before you build anything.  I'm sure people will disagree with this one, but to me its VERY important when you have flood plains, to drop a hamlet on every single flood plain.  If the city has 5 or 6... I would build them on every single one.  It has the 3 food production anyways (same as farmed grassland) and since its on a stream, it has 1 gold per turn before you build.  As the game progresses, that city will be awesome on money production and science as the hamlets grow and as banks, groceries and markets are produced.
2. Religion.  Again, not as 'fun' as crushing your neighbor with unique units and things like that, but starting with a Spiritual civ that has a head start on Mysticism can get you a head start on founding a religion.  Why would you do that?  "Money."  Think "money..." this isn't CIV III.    Assuming you get a great prophet in the game somewhere (I get at least a few on avg. per game) you use the great prophet to create your religion's special building in the city that founded the religion.  That gives you 1 gold per city.  Doesn't sound like a lot maybe, but if you have 10-20 cities, and you spread it like wildfire to your neighbors, that one building can be the difference in 20% or so science with that alone.  Something to think about.... your neighbors will dislike you if they have a different religion, so spam them with missionaries.  I like creating Buddhism with a Spiritual civ and spam your neighbors with missionaries early when they have no religion.  They will become Buddhist which equals more money later to you, and they will be friendly to you because of your same religion. Also important is that when you kick their butt later in the game you can see into every city of theirs to see all of their units (when you have built the unique building to that religion with a Prophet).  Nice, huh?  Works with every religion, but I'd recommend going after an early one.  One more suggestion.  Expand your religion to each of your cities and switch to Organized Religion to gain that 25% building bonus.  Think of it as an extra factory just because of your faith.
3. The order that you study technologies.  If you get Iron Working or something like that, sure, its nice and has benefits, but usually about that point in the tech tree, is when you're expanding to the point of hurting for money and progressively turning down the science slider.  I would suggest getting the 1st couple of rows of techs for defense and offense, but rather than going for horseback riding or iron working and things around there... consider going a little unbalanced on the tech tree to get Currency early.  Building markets early helps keep science high, especially if you followed suggestion #1 above on making Hamlets.  Yes, the war-monger in me hates taking several turns to make markets on all my civs, but then you can unleash the inner war monger and afford war.  I do the same with Banking as I usually go a little unbalanced to get that and build banks everywhere.  If you're a little behind on techs, going unbalanced and then trading for the older tech's isn't a bad idea. Though I don't like trading techs, it can be necessary.

I played a game with the Indians a couple of times where I NEVER dropped the science slider below 100%.  In one of those games, I founded every religion and had the special building for almost every religion.  I spammed everyone with missionaries and was wealthy beyond belief.  I bought buildings, units, etc with the spare gold and had it rolling in.  Remember that even 'just' 20 cities of 1-gold per turn for them having your religion is boosted by banks, markets and groceries as well, so it helps more than *just* the gold per turn from each religion/city.

Keep on top of courthouses and forbidden palace if you want to save money, especially if your empire is big and cities are taking high maintenance from distance.  Think 'money!'  Think long-term money.

Last strategy.  If you think it'll take you 3 axemen to capture a city, take 6.  I've seen the automated CIV suggestions when you're loading a game tell you not to take 50 when 15 will do or something along those lines.  I think its a great idea to take more than you need and always take around double what I think when I'm picking a fight.  Poor 'dice-roll' luck or unexpected surprises make it nice to have twice the number you think you'll need.  Then you can take them and roll on to the next city.  Also, pick a fight early once money is taken care of. 

These are especially effective on the bigger maps and slower games.

As always, people will disagree, but these are my thoughts and suggestions.  Please respond as you see fit. 

It took a while to learn that you can't build cities as often as you would like, when in past versions, you couldn't get them founded often enough. It definitely takes quite a bit of getting used to.

I'm still learning the game myself, and everything you have said above makes perfect sense to me... still getting used to not building cities as much as Civ3...

The jump from Civ III to cIV was pretty different (putting it lightly) and it would've been much easier if there were guides like this back then.