Do different databases use different name quote?

This use of quotes is called delimited identifiers. It's an important part of SQL because otherwise you can't use identifiers (e.g. table names and column names) that:

• Include whitespace: "my table"
• Include special characters and punctuation: "my-table"
• Include international characters: "私のテーブル"
• Are case-sensitive: "MyTable"
• Match SQL keywords: "table"

The standard SQL language uses double-quotes for delimited identifiers:

SELECT * FROM "my table";

MySQL uses back-quotes by default. MySQL can use standard double-quotes:

SELECT * FROM `my table`;
SET SQL_MODE=ANSI_QUOTES;
SELECT * FROM "my table";

Microsoft SQL Server and Sybase uses brackets by default. They can both use standard double-quotes this way:

SELECT * FROM [my table];
SET QUOTED_IDENTIFIER ON;
SELECT * FROM "my table";

InterBase and Firebird need to set the SQL dialect to 3 to support delimited identifiers.

Most other brands of database use double-quotes correctly.

SQL Server uses [square brackets] or "double quotes" when QUOTED_IDENTIFIER option is ON.

I believe double quotes are in the SQL-92 standard.

Succinctly, yes.

The SQL standard uses double quotes around the name to indicate a 'delimited identifier'.

Informix by default uses single and double quotes interchangeably to indicate character strings. However, by setting the environment variable DELIMIDENT you can turn on the SQL standard behaviour - single quotes around strings and double quotes around delimited identifiers.

Other people have listed other behaviours for other DBMS; I don't need to repeat those.