While studying relational databases, I ran into this confusing page where the following snapshot summarizes my confusion:
Why does the author say that Subject Table is in 1NF when student_id 401 and subject Math is repeated in the same way the blue depicts? This seems like a contradiction.
The key concept this page fails to mention is that the data contained in the row must not be duplicated, meaning that a single key value cannot have multiple values for a single column. The real problem is that for student id of 401, the first table specifies the name twice.
The text following the last table should read "In Subject table concatenation of subject_id and student_id is the primary key". The reason the new subject table is okay is because the key is actually both of these values, so while 401 is repeated, the key is 10, 401 and 11, 401, which are two distinct values. Likewise, math is repeated, but it is the data for two separate keys; it isn't repeated within a row. Because these key values are different, it is okay.
The page you referenced doesn't give a very precise definition. I hope this explanation helps. Keep checking other sites to get a more clear understanding. Wikipedia has a good example and a precise definition, though it is a bit abstract and hard to follow. http://en.wikipedia.org/wiki/First_normal_form