For a profile that you manage - simply select edit among the tabs (assuming you are logged in).
For profiles managed by others, depending on how contemporary the person's life is in the record, there are degrees of privacy or security and you would need to be added to a profile's trusted list to edit one that is more recent.
Some profiles also have project protected status - which means again you would need to be added to the trusted list, and it presumes you have an understanding of the project they fall under.
There are two main parts you can edit - those that have fields, and the general biography section.
Happily this is a wiki, so don't worry too much that anything will be lost!