In relational database theory, normalization is the process of restructuring the logical data model of a database to eliminate redundancy, organize data efficiently, and reduce the potential for anomalies during data operations to improve data consistency and future enhancements

A non-normalized database can suffer from data anomalies:

A non-normalized database may store data representing a particular referent in multiple locations. An update to such data in some but not all of those locations results in an update anomaly, yielding inconsistent data. A normalized database prevents such an anomaly by storing such data (i.e. data other than primary keys) in only one location.
A non-normalized database may have inappropriate dependencies, i.e. relationships between data with no functional dependencies. Adding data to such a database may require first adding the unrelated dependency. A normalized database prevents such insertion anomalies by ensuring that database relations mirror functional dependencies.
Similarly, such dependencies in non-normalized databases can hinder deletion. That is, deleting data from such databases may require deleting data from the inappropriate dependency. A normalized database prevents such deletion anomalies by ensuring that all records are uniquely identifiable and contain no extraneous information.
Normalized databases have a design that reflects the true dependencies between tracked quantities, allowing quick updates to data with little risk of introducing inconsistencies. Instead of attempting to lump all information into one table, data is spread out logically into many tables. Normalizing the data is decomposing a single relation into a set of smaller relations which satisfy the constraints of the original relation. Redundancy can be solved by decomposing the tables. However certain new problems are caused by decomposition. Normalization helps us to make a conscious decision to avoid redundancy keeping the pros and cons in mind.