A new modern computer will be much quieter than its equivalent from 10 years ago. This is partly due to temperature-controlled fans that only speed up if the computer starts to get hotter, and partly due to better design of the fans. In particular, modern fans are usually bigger, allowing them to spin more slowly and therefore make less noise.
However, as your new computer gets older, you may notice a fan becoming noisier. It might be spinning at full speed more often than it used to, or it might be making a whining or grinding noise. Fortunately, while annoying, these problems are rarely serious as long as they are not left for too long.
The usual cause of whining or grinding noises is that the bearing in the fan is drying out. This may be particularly noticeable when the computer is first turned on and the remaining oil in the bearing is cold. While it is possible to add oil to the fan, this will only be a short-term fix as it will quickly dry out again. The permanent solution is to replace the fan.
When a fan is noisy because it is regularly running at a fast speed, this is usually due to some sort of contamination - dust, hair, fibres, cigarette smoke or, if the computer is regularly used in a kitchen, contamination from cooking.
Dust, hair and fibres are common problems in any domestic situation. Long-haired pets can be a particular problem because the hairs they shed can get into the computer, where they end up insulating components which then get too hot. New carpets are another common problem, as they tend to release dust and fibres for some time after they are laid. The dust and fibres coat and clog up the cooling blocks (known as heat sinks) in the computer, so they become less efficient at removing heat, causing the fan to have to work harder. Cigarette smoke makes these problems much worse - nicotine is sticky and will coat the inside of the computer, causing other contaminants to stick to every surface, causing overheating.
The usual solution to these problems is to disassemble the computer and clean it! A can of compressed air is often all that is required (although it will make a mess, blowing the dust everywhere). With a desktop computer, you can do this yourself, although you must take precautions against electrostatic discharge, which can damage sensitive electronic components. Laptop computers can be much more fiddly and the plastics they are made of can be quite brittle and easily broken. This is a case where repair is relatively quick, so if you don't want to tackle it yourself, expert help will not be expensive.