Why use a 1-to-1 relationship in database design?

From the logical standpoint, a 1:1 relationship should always be merged into a single table.

On the other hand, there may be physical considerations for such "vertical partitioning" or "row splitting", especially if you know you'll access some columns more frequently or in different pattern than the others, for example:

• You might want to cluster or partition the two "endpoint" tables of a 1:1 relationship differently.
• If your DBMS allows it, you might want to put them on different physical disks (e.g. more performance-critical on an SSD and the other on a cheap HDD).
• You have measured the effect on caching and you want to make sure the "hot" columns are kept in cache, without "cold" columns "polluting" it.
• You need a concurrency behavior (such as locking) that is "narrower" than the whole row. This is highly DBMS-specific.
• You need different security on different columns, but your DBMS does not support column-level permissions.
• Triggers are typically table-specific. While you can theoretically have just one table and have the trigger ignore the "wrong half" of the row, some databases may impose additional limits on what a trigger can and cannot do. For example, Oracle doesn't let you modify the so called "mutating" table from a row-level trigger - by having separate tables, only one of them may be mutating so you can still modify the other from your trigger (but there are other ways to work-around that).

Databases are very good at manipulating the data, so I wouldn't split the table just for the update performance, unless you have performed the actual benchmarks on representative amounts of data and concluded the performance difference is actually there and significant enough (e.g. to offset the increased need for JOINing).

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On the other hand, if you are talking about "1:0 or 1" (and not a true 1:1), this is a different question entirely, deserving a different answer...

See also: When I should use one to one relationship?

Separation of duties and abstraction of database tables.

If I have a user and I design the system for each user to have an address, but then I change the system, all I have to do is add a new record to the Address table instead of adding a brand new table and migrating the data.

EDIT

Currently right now if you wanted to have a person record and each person had exactly one address record, then you could have a 1-to-1 relationship between a Person table and an Address table or you could just have a Person table that also had the columns for the address.

In the future maybe you made the decision to allow a person to have multiple addresses. You would not have to change your database structure in the 1-to-1 relationship scenario, you only have to change how you handle the data coming back to you. However, in the single table structure you would have to create a new table and migrate the address data to the new table in order to create a best practice 1-to-many relationship database structure.