Export db, change charset, convert and import again:

mysqldump -u username -p --add-drop-table database-name | replace CHARSET=latin1 CHARSET=utf8 | iconv -f latin1 -t utf8 | mysql -u username -p -D database-name

Directly via mysql, note utf8 charset is not fully compatible with UTF-8, so better use utf8mb4:

#change default only
ALTER TABLE tablename CHARACTER SET utf8mb4 COLLATE utf8mb4_unicode_ci;
#convert charset of data
ALTER TABLE tablename convert to character set utf8mb4 COLLATE utf8mb4_unicode_ci;

#changes for whole database
ALTER DATABASE databasename CHARACTER SET utf8mb4 COLLATE utf8mb4_unicode_ci;
ALTER DATABASE databasename convert to character set utf8mb4 COLLATE utf8mb4_unicode_ci;

For websites, ensure that the header and document type are set correctly:

header('Content-Type: text/html; charset=UTF8');

<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />

php.ini should have:

; PHP's built-in default is text/html
default_mimetype = "text/html"
default_charset = "utf-8"

my.cnf should have:

[client]
default-character-set=utf8
[myslqd]
skip-character-set-client-handshake
collation_server=utf8_unicode_ci
character_set_server = utf8 

alternatively, in php scripts set:

@mysql_query("SET NAMES utf8") or die ("Could not set character set to UTF8: ".mysql_error());