If you have written a full ensemble score, MuseScore can create sheet music that shows only the individual part for each musician in the ensemble.
Note: In the current version of MuseScore, it is not possible to split a single staff (that contains two or more voices) into separate parts. So, any instrument that you want to print out a separate part for also needs to have its own staff in the full score.
Set up all parts at once
If you have an orchestral score in which each instrument should get its own part, you can create the parts more easily:
Go to File → Parts...
Click the New All button (parts are named with the instrument name, and if needed with an appended "-<number>")
Now you will see a tab for each part with your score.
Define specific parts
You can define the parts at any point after creating a new score. You only need to define the parts once for each score, but you can make changes if needed. The following instructions use a string quartet as an example, but the same principles apply for any other ensemble.
From the main menu, choose File → Parts...
In the Parts window click New to create a "part definition"
In the right pane, type the words you want to use for the "Part title" (this also serves for the corresponding part of the filename when exporting)
Pick the instrument that you want to appear in your part by marking the relevant box in the right-hand pane. Usually, you only want one instrument per part, but sometimes you might need a part that includes more than one instrument (such as multiple percussion staves). MuseScore allows you to mark as many instruments per part as you need
Repeat steps two through four (above) for each part as needed
Once you're done, click OK to dismiss the Parts window
You have now finished setting up the parts. You do not need to do this again, unless you add or remove an instrument from your full score.
Exporting the parts
Go to File → Export Parts...
Navigate to the place you want them to be exported to and select the file format (PDF is the default)
For filename just enter whatever prefix is useful for all parts, or leave the default (the filename of your score)
This will generate files with the names "<title>" + "-" + "<part name>.<extension>". In addition, when exporting as PDF, this will also generate "<title>" + "-Score_And_Parts.pdf".
Parts and score are "linked", which means that any change to the content in one will affect the other, but changes to the layout will not. When you have the parts created, they are saved along with the score (if you open the score you have tabs for the score and every part you created).
You can, however, save each part individually by selecting its tab and use File → Save As...