How important is your score in Civilization IV?

Score can be a good barometer of how well you're playing, but it's only a factor in winning if you run the clock completely out and nobody wins another way.

It's sometimes easy to "back door" win via one of the other victory conditions even if you're behind in score, but being the clear score leader makes every victory easier. Higher score generally means you're more technologically advanced and larger in terms of population and number of cities. All of these are important factors to every victory type.

Extended war with no real significant movement of borders is a bad way to win no matter how you're playing. Every unit lost is shields that are being thrown away. If you're going to devote time and resources to a war, make sure you've got an achievable goal, and that you're not neglecting your technological and economic progress.

Technological advances and a strong economy are as important, if not more important, than the size of your army. If you can't keep your technology current and your finances significantly in the black, don't go to war or try to end wars you're in quickly.

If you're new to the series, I'd suggest learning to focus your cities on a particular type of output. For instance, for food/population you'll want to focus on building near rivers and plains. Your first city will likely be in an area like this.

Your second and third cities should be focused either on generating money/science (plains/grassland, primarily) or production (hills and forest). Only build buildings and terrain improvements that focus on the city's specialization. The "production" city should focus on building military improvements and military units. Improve terrain with mines and lumber mills. The economic/technology cities should focus on boosting trade income with Cottages.

I'm going to take your question as meaning "how good is my score as an indicator of how well I'm doing so far and likely I am to win". If that wasn't the intention plesae feel free to ignore.

As far as I can tell, the answer is "only somewhat". This is solely based on experience and observation. On the lower levels I find that my score tends to start high (relative to other civs) and stay that way, as long as I'm playing well. On higher levels though I find my score is always low at the start, and climbs up only relatively slowly. As far as I can tell this is because of the bonusses that AI civs get at higher levels. It's also because you need to start making decisions that will pay off in the long term rather than the short term at higher levels.

This opinion is also supported by the fact that I see other civs relative score fluctuate quite a lot, especially during the mid-game. I've seen them go from top to bottom, even without a war.

Scores reflect current position, not potential. making a really good tech trade can bump you many places in the ranking. But building a whole lot of universities won't do that - you need to wait for the techs you can research because of them to appear.

the score is one of the victory conditions of Civilization. Many things affect this, basically, your territory, your technology, your cities, your wonders, your army, etc.
If the victory condition is global domination then score doesn't mean anything, as you'll have to conquer most of the planet to win anyway, and if you do it, chances are high that your score will also be the highest. But if you remove that victory condition, then yes, it's pretty important.
It also gives you an idea of how likely you're to get creamed if you attack a new civilization you meet :)