Search a whole table in mySQL for a string

Identify all the fields that could be related to your search and then use a query like:

SELECT * FROM clients
WHERE field1 LIKE '%Mary%'
   OR field2 LIKE '%Mary%'
   OR field3 LIKE '%Mary%'
   OR field4 LIKE '%Mary%'
   ....
   (do that for each field you want to check)

Using LIKE '%Mary%' instead of = 'Mary' will look for the fields that contains someCaracters + 'Mary' + someCaracters.

Try something like this:

SELECT * FROM clients WHERE CONCAT(field1, '', field2, '', fieldn) LIKE "%Mary%"

You may want to see SQL docs for additional information on string operators and regular expressions.

Edit: There may be some issues with NULL fields, so just in case you may want to use IFNULL(field_i, '') instead of just field_i

Case sensitivity: You can use case insensitive collation or something like this:

... WHERE LOWER(CONCAT(...)) LIKE LOWER("%Mary%")

Just search all field: I believe there is no way to make an SQL-query that will search through all field without explicitly declaring field to search in. The reason is there is a theory of relational databases and strict rules for manipulating relational data (something like relational algebra or codd algebra; these are what SQL is from), and theory doesn't allow things such as "just search all fields". Of course actual behaviour depends on vendor's concrete realisation. But in common case it is not possible. To make sure, check SELECT operator syntax (WHERE section, to be precise).