The terms Priority & Severity are used in bug tracking to share the importance of the bug among the team & to fix it.

Priority:

The priority status set by the team lead & he is consulting with client as per project requirement.
Priority terms represents the importance of fixing a bug & how soon bug should be fixed. If priority is high mentioned then developer has to fix bug at the earliest. If priority is low mentioned then developer has to it at next build.
For example : If the company name is misspelled in the home page title of the website, then priority is high & severity is low to fix the bug, It affects the bad impression on clients if company name is incorrect on home page so priority status is high in this case.

Priority can be of following types:

Show-stopper - this types of bug needs to be fixed instantly, the build cannot be released with this defect.
Urgent - needs to be fixed before any other high, medium or low defect should be fixed.>
High - should be fixed as early as possible.
Medium - should take precedence over low priority defects.
Low - fixing can be deferred until all other priority defects are fixed.

Severity:

Severity status set by tester. Severity is how badly the bug is impacting the application.
For example, If an Application crashes while using the some feature of xyz functionality of application when used with non recommended browser lat say(IE7) severity wise defect is high as it cause the system to crash, but priority wise it is mark as low priority as it as been checked by non recommended browser.

Severity can be of following types:

Critical: This type of bug happens when it causing system failure. Nothing can proceed further.
Major: This type of defect is highly severe defect, is causing the system to collapse but few parts or feature of the app is still usable.
Average: is causing some undesirable behavior, however system / feature is still usable to a high degree
Minor: This type of severity issues are more of a cosmetic issues. No serious impedance to system functionality is noted