Imagine you have to describe many aspect of the same thing. You have to choose which aspects you want to take care of: these are the columns of your table. For each of the column you can choose the data type to represent (numeric, String, ...). The column could be also a composite data type (ex: date) or a reference to another object.
The description of an object consists in all the values contained in the columns relative to that object: this is the row/record (the two terms have the same meaning in ER databases).
The field is the value assumed by a column--let's say that it is a cell in a table. It is part of the row, but it may have no sense outside the context provided by the row and the columns.
Maybe the confusion is due to the fact that to simplify the notation, the term field is used as the term column. When you see a query like "select * from foo where somefield=something" it means that you select the rows in which the field relative to the column "somefield", assumes the value "something".
This definition is ok also with HTML fields. A field of a form is the place where you enter the value the column will have in your case, that means, in the row that represents you.