Separating paragraphs with blank lines is easiest for readers to scan, but they can also be separated by indenting their first lines. This is often used to take up less space, such as to save paper in print.
Writing that is intended to be edited, such as school papers and rough drafts, uses both blank lines and indentation for separation. In finished works, combining both is considered redundant and amateurish.
In very old writing, paragraphs were separated with a special character: ¶, the pilcrow. Nowadays, this is considered claustrophobic and hard to read.
How hard to read? See for yourself: