Any non-trivial data model has relationships and databases and application platforms may or may not support the notion of storing the relationships inline, that is, denormalizing the relationships into the main record. To give a real example, a user may have a set of roles in the application and this user-role relationship can be either stored in a separate table or directly within the user table. However, since the user-role relationship is 1-to-many, storing this directly within the user data model requires storing the relationship as a multi-valued column and arrays are perfect for that.
The challenge that does come with storing the 1-to-many relationsips as arrays is that it then makes it difficult to do certain SQL queries that are easy to do had the relationship been stored explicitly. That is why some databases offer a special operator to deal with such array fields.