Both words seem to mean the following:
Having or feeling no interest in something or someone.
When do we use disinterested vs. uninterested? Are they interchangeable?
The two words are used very differently in standard English. Both mean that the person or thing being spoken of has no interest. However, the word interest has two different but related meanings. (There's also monetary interest, which is basically rent paid for use of money, but that's not relevant to the question.)
In one sense, interest means that which appeals to a person, or that which they enjoy. Uninterested would therefore be used to mean that someone is bored or apathetic:
A person who does not enjoy watching sports could be correctly described as "uninterested in sports".
The second meaning for interest is "to have a stake in", or "in some way to have one's life be affected by an outcome". Someone who places a wager on the outcome of a football match has an interest in the result.
Because such interest can be problematic, a trusted person who doesn't have any reason to care about an outcome is said to be disinterested. Think of a referee at a sporting event, or an attorney who helps draw up a contract. These people get paid regardless of the outcome. They may not be personally uninterested in what happens, but they are professionally disinterested because that is what they are paid for.
As Eric Lippert comments, disinterested is sometimes used in a context where uninterested would be more appropriate. This isn't out of any sense of correctness, but merely comes from the modern trend of picking a word that sounds right without bothering to find out whether it actually fits. (In the same way that infer is often used where imply would be more correct.) Thus, a usage like this might be understood, but should be avoided:
Macmillan Dictionary gives two senses for the definition of "disinterested". The second one is "not interested", with a note "Many people think that this use of the word is not correct, and prefer to use uninterested." ("Uninterested" is simply defined as "not interested".)
The following is the full definition: