On some fonts, like Times Roman, you may notice that there are small embellishments on each letter. On the legs of a capital 'A' for instance, there are decorations commonly known as serifs. They come from the the flourishes and marks created when you write letters by hand: depending on your pen's angle, there are different line widths and marks. Those marks were considered a part of the letterform when older typefaces were designed, which have evolved into the many serif fonts available today.
In contrast, there are also sans serif fonts — which deliberately leave off serifs, as the typeface Helvetica does. Deciding when to use a sans serif font, as opposed to a serif font, can be an aesthetic decision, but it can also be a question of legibility. In many cases, large text blocks are considered to be easier to read when set in a serif font, while sans serif fonts are considered ideal for headlines and signage.