Customer Portal is a part of the Salesforce.com Service Cloud product suite where a portal can be made available to your customer community where they can log/view cases, access knowledge base and also offers you the flexibility to expose custom objects via the customer portal. You cannot expose Accounts, Contacts, Leads, Opportunities etc. on the Customer portal. Customer portal is mailnly used for Customer Support as a self-service support channel. When an Account and Contact are created in the system, you can then enable the Contact as a Customer portal user in the system and he/she will get login instructions to login to the Customer portal.

Partner Portal is a part of the Salesforce.com Sales Cloud, mainly related to Partner Relationship Management processes. This portal is applicable in Channel sales model where companies sell via their partners and distributors. When Partner portal is enabled, Partner users are able to access Accounts, Contacts, Opportunities, Leads, Cases etc. via the Partner portal. Idea here is to extend your CRM data to your partners in a restricted mode so that you can work more closely and effectively with your partners on closing deals and logging leads. When an Account and Contact are created in the system, you can then enable the Contact as a Partner portal user in the system and he/she will get login instructions to login to the Partner portal.

Also, please note that Customer and Partner portal are now replaced with Customer Community and Partner Community with extended capabilities like Chatter can now be exposed to Customer and Partners via the Community portals. More info is available at http://www.salesforce.com/chatter/features/communities/

Hope this helps!