How to insert a value that contains an apostrophe (single quote)?

Escape the apostrophe (i.e. double-up the single quote character) in your SQL:

INSERT INTO Person
    (First, Last)
VALUES
    ('Joe', 'O''Brien')
              /\
          right here  

The same applies to SELECT queries:

SELECT First, Last FROM Person WHERE Last = 'O''Brien'

---

The apostrophe, or single quote, is a special character in SQL that specifies the beginning and end of string data. This means that to use it as part of your literal string data you need to escape the special character. With a single quote this is typically accomplished by doubling your quote. (Two single quote characters, not double-quote instead of a single quote.)

Note: You should only ever worry about this issue when you manually edit data via a raw SQL interface since writing queries outside of development and testing should be a rare occurrence. In code there are techniques and frameworks (depending on your stack) that take care of escaping special characters, SQL injection, etc.

You just have to double up on the single quotes...

insert into Person (First, Last)
values ('Joe', 'O''Brien')

You need to escape the apostrophe. In T-SQL this is with a double apostrophe, so your insert statement becomes:

Insert into Person
(First, Last)
Values
'Joe', 'O''Brien'