Like the name suggests, relational data are items that can be related to one another through the use of a key value. For example, your government issued an identifier number to you that is used to identify you. This number is then linked to your name, where you were, born, etc. It is a key to reaching that information.
If you think of a table of information, with rows running horizontally across the page and columns intersecting the rows vertically, running down the page, the key value identifies the row. So, for information about person you might have something like this:
Each key identifies a row, and each row has columns of information. Then, in another table where you store, say, order information, you do not have to enter John Smith's personal information every time you create an order for him. The data exists in one place. Then, if you find out that he spells his name "Jon" and not "John" you can change it in once place.
There are other advantages to the model, but this is the crux of it. Information only has to be entered once, and then related to other information.
A relational database management system (RDBMS) is software that allows you manipulate this type of information easily.
Non-relational is a grab bag of different approaches that do not include necessarily the idea of a key value (or at least not in a separate tables sort of way). Generally a non-relational data store will store all of the information for any one thing all together, all at once. For example, an order would be stored all on one line, with the customer information repeated over and over for each order they are linked to. You wind up storing the information more than once for certain items and you have a lot of work to do if you want to update that information or use it to link to information that is held elsewhere.
Storing information in a spreadsheet often winds up being non-relational but in more modern systems there are ways to make the data relate to other data. The lack of management tools, however, will generally drive you towards a management system.
If the data you are storing is VERY simple and small in size you can save time and money by just storing it into a text file that you update by hand. For anything beyond that you should consider more sophisticated software to help you with the process of storing and updating the information.
Weighing the pros and cons of the various different packages is a very involved process. Think through what your needs will be and then invest the time in some research. Make sure you base your decision on your needs and not on what some marketing tells you your needs are.
Make a techgnostic choice for your database needs and you will achieve your goals effectively and efficiently.