Few important things about the TABS are:
Tabs can help you in making the views, which help user to see the information at a glance. Example: List of all contacts whose birthday is in this month, list of all opportunities which are in negotiation stage.
There is a limitation on the number of tabs one can have in their organization, Maximum of 25 tabs are allowed in the enterprise edition.
For a profile you can set the status of tabs as "Default On", "Default Off" and "Tab Hidden"
Default On: User with this Tab setting will be able to see the Tab for custom object every time he enters in to the SFDC.
Default Off: User with this Tab setting will NOT be able to see the Tab for a custom object when he enters in to SFDC but has the ability to add this tab to his view by customizing the tab view.
Tab Hidden: User with this Tab setting will be NOT be able to see the Tab for a custom object and even will NOT be able to bring the tab by customizing the tab view settings.
Search is enabled only for those custom objects that have a Tab. So if you are unable to search on a custom object then the reason could be that the Tab for that custom object is not enabled in profiles or you have not created the Tab for that object. DEV 401 Certification Question
Web Tabs are the tabs from which user goes to the external websites. So use them when you want to divert the user from SFDC to External Systems.