There are a couple of ways where you can lose money. On the F1 Domestic Advisor screen, you can see a breakdown of where you are losing your money (as well as where you are raking in the cash). Here's a description of what those lines mean:
Science: This is pretty straightforward. It's how much gold is being converted into beakers for science research. The higher your science slider is (the slider to the right with the beaker), the more money you will spend on science instead of earning in gold. This is an easy way of earning more gold - turn down the slider and sacrifice some science for gold, if you are losing lots of gold. Also, when you only have 1 turn left to research a tech, try turning the science slider down - sometimes you can still get the tech in 1 turn, while earning extra gold.
Entertainment: This represents another slider - the luxury tax slider (below the science slider). It represents how much gold is being turned into "happy faces", or entertainment to turn your citizens in cities happy to prevent riots (a riot happens when >50% of the citizens in a city are unhappy). This you usually can't turn down without having some repurcussions in your cities.
Corruption: This shows how much money is being lost to corruption. Every city doesn't utilize every commerce it earns - some is just, literally, 'lost', as in you can't use it. This is usually higher the larger your empire is, as there will be more corruption. For ways to lower this, search the forums.
Maintenance: This is how much gold you are paying to maintain city improvements. For example, if you have 5 cities, and each has a temple (1 gold in upkeep per temple), you will pay 5 gpt in upkeep. This can add up quickly with a large empire, and it is hard to stop; you have to watch what you build and make sure it is necessary.
Unit Costs: This shows how much money is being paid to maintain units that aren't supported by your government. Different governments have different limits, and how much you pay in upkeep. For example, if you go over the limit in Monarchy, you pay 1gpt per unit extra. Republic is 2gpt. Feudalism is 3gpt! Make sure you don't go too high over the limit for this, as it can add up quickly. It helps to disband old units or units in your core away from borders of other civs.
To Other Civs: This is pretty simple - it is how much gold per turn you are giving to other civilizations in diplomacy (e.g. If you trade 10gpt to a civilization for a technology, "-10" will appear in this column. Similarly, there is one on the positive side of things for money you are receiving.
Here are some good ways to earn some money besides the one mentioned above: try building marketplaces, banks, and stock exchanges - they increase gold by 50%. Roads are also the key. Every tile with a road produces an extra commerce. The governments Republic and Democracy all have a trade bonus so that for every tile that produces one commerce, you'll get an extra. That is one reason Republic is considered the best government. Lowering corruption is an effective way of earning money too.