Show constraints on tables command

Simply query the INFORMATION_SCHEMA:

USE INFORMATION_SCHEMA;
SELECT TABLE_NAME,
       COLUMN_NAME,
       CONSTRAINT_NAME,
       REFERENCED_TABLE_NAME,
       REFERENCED_COLUMN_NAME
FROM KEY_COLUMN_USAGE
WHERE TABLE_SCHEMA = "<your_database_name>" 
      AND TABLE_NAME = "<your_table_name>" 
      AND REFERENCED_COLUMN_NAME IS NOT NULL;

afaik to make a request to information_schema you need privileges. If you need simple list of keys you can use this command:

SHOW INDEXES IN <tablename>

The main problem with the validated answer is you'll have to parse the output to get the informations. Here is a query allowing you to get them in a more usable manner :

SELECT cols.TABLE_NAME, cols.COLUMN_NAME, cols.ORDINAL_POSITION,
cols.COLUMN_DEFAULT, cols.IS_NULLABLE, cols.DATA_TYPE,
    cols.CHARACTER_MAXIMUM_LENGTH, cols.CHARACTER_OCTET_LENGTH,
    cols.NUMERIC_PRECISION, cols.NUMERIC_SCALE,
    cols.COLUMN_TYPE, cols.COLUMN_KEY, cols.EXTRA,
    cols.COLUMN_COMMENT, refs.REFERENCED_TABLE_NAME, refs.REFERENCED_COLUMN_NAME,
    cRefs.UPDATE_RULE, cRefs.DELETE_RULE,
    links.TABLE_NAME, links.COLUMN_NAME,
    cLinks.UPDATE_RULE, cLinks.DELETE_RULE
FROM INFORMATION_SCHEMA.`COLUMNS` as cols
LEFT JOIN INFORMATION_SCHEMA.`KEY_COLUMN_USAGE` AS refs
ON refs.TABLE_SCHEMA=cols.TABLE_SCHEMA
    AND refs.REFERENCED_TABLE_SCHEMA=cols.TABLE_SCHEMA
    AND refs.TABLE_NAME=cols.TABLE_NAME
    AND refs.COLUMN_NAME=cols.COLUMN_NAME
LEFT JOIN INFORMATION_SCHEMA.REFERENTIAL_CONSTRAINTS AS cRefs
ON cRefs.CONSTRAINT_SCHEMA=cols.TABLE_SCHEMA
    AND cRefs.CONSTRAINT_NAME=refs.CONSTRAINT_NAME
LEFT JOIN INFORMATION_SCHEMA.`KEY_COLUMN_USAGE` AS links
ON links.TABLE_SCHEMA=cols.TABLE_SCHEMA
    AND links.REFERENCED_TABLE_SCHEMA=cols.TABLE_SCHEMA
    AND links.REFERENCED_TABLE_NAME=cols.TABLE_NAME
    AND links.REFERENCED_COLUMN_NAME=cols.COLUMN_NAME
LEFT JOIN INFORMATION_SCHEMA.REFERENTIAL_CONSTRAINTS AS cLinks
ON cLinks.CONSTRAINT_SCHEMA=cols.TABLE_SCHEMA
    AND cLinks.CONSTRAINT_NAME=links.CONSTRAINT_NAME
WHERE cols.TABLE_SCHEMA=DATABASE()
    AND cols.TABLE_NAME="table"

I use

SHOW CREATE TABLE mytable;

This shows you the SQL statement necessary to receate mytable in its current form. You can see all the columns and their types (like DESC) but it also shows you constraint information (and table type, charset, etc.).