If you split white light in a prism you will get a spectrum of colours - Red, orange yellow, green, blue, indigo, violet - and you will see that they merge into each other. Each colour has a different wave length and as wavelength is a continuous (as in not discrete) variable then you could argue that there are an infinite number of colours.
To confuse you even more we can also see colours because they are a mix of different wavelengths. Your computer screen can only produce the colours red, green and blue but it produces them with different intensities to produce a certan colour.
You also need to remember that that is just what healthy human eyes see - other species can see wavelengths beyond what we can.