I dumped a shop from my customer and imported it to my local system, as you can guess, the local domain is another one. All my domains are like: <customer>.dev.

Where to change domains

Secure/Unsecure Base URL

From core_config_data:

default	0	web/unsecure/base_url	http://customer.dev/
default	0	web/secure/base_url		http://customer.dev/

Check the skin/media/js url although!

default	0	web/unsecure/base_link_url	{{unsecure_base_url}}
default	0	web/unsecure/base_skin_url	{{unsecure_base_url}}skin/
default	0	web/unsecure/base_media_url	{{unsecure_base_url}}media/
default	0	web/unsecure/base_js_url	{{unsecure_base_url}}js/
default	0	web/secure/base_link_url	{{secure_base_url}}
default	0	web/secure/base_skin_url	{{secure_base_url}}skin/
default	0	web/secure/base_media_url	{{secure_base_url}}media/
default	0	web/secure/base_js_url		{{secure_base_url}}js/

Cookies

For cookies you should add the domain and the path, to secure your cookie. In your development environment, you can just use these settings:

default	0	web/cookie/cookie_path		NULL
default	0	web/cookie/cookie_domain	NULL