For example an application which generates some banking 
related reports weekly, monthly, quarterly & yearly by 
doing some calculations. If there is a error while 
calculating yearly report. This is a high severity but low 
priority because this error can be fixed in the next 
release as a change request.

EX:1If the logo is not displaying on the front page
EX2: if there there is misspell in th url like instead of 
google it displayed as gogle

In this situations Low severity and  high priority

Severity : Means impact on the application..,
Priority : Means Importance in terms of both application 
           and client

Example: 
Assume that Login page.., Consits of User Name and 
Password.., crietria is that the Password as to expire in 
every 30 days and new password has to be assigned..,old one 
should no longer exist

For High Severity and low Priority
1. Application accpets the New password after very expiry 
of 30 days.., Now the Application still accpets the Old 
password and the new password for the same account..,
So hence this High severity and Low severity .., since the 
deveats the criteria.., ie Old password still works..

For High Priority and low Severity
2. Application behaves such that it doesnt allow to change 
every 30 dyas.., it still works with same old password
hnece deviates from the clinest requirements and doesnt 
affect the application.., hence it is tersmed as High 
Priority and low severity

Severity: How bad a bug is? Means impact on the application
Priority: How soon the bug need to be fixes? Means 
Importance in terms of both application and client

Example: 
Let us take an example of data entry screen of an 
application.

For High Severity and low Priority
1. Application get closed itself after entering 100 records 
then again needs the application to restart. 

For High Priority and low Severity
2. After entering record the message prompt showing the 
message has some spell mistake.

Vimal (Test Engg.)

Severity : How severe the defect is.

Priority : It is a sequence of defects that tells that what
defect is first resolved and what next.

High severity and low priority : If the test Engg. raise a
defect that is a related to out of scope functionality even
if it is a high severity defect u assign low priority to
that defect.

Low severity and high priority : The defects that are
related to Look and Feel and Spelling mistakes that have low
severity but in customer perception that defect has high
priority.

Correct me if u feel wrong.

1.Low Severity and High priority:-
Let us discuss with a simple login screen.

 Enter username and enter password and press login button,
if it is not accepting it then it would be difficult for
end-users to work with the application.For this scenario the
 impact would be very high like lot of end-user may not
complete there task. which is high priority but solving them
is really simple just go to database and check the login
connection and re-set the password which takes only few min
to solve but until solved it is stopping lot of end-user to
complete there task. so priority is high ( to be resolved
immediately but resolving is simple so low severity).

High Severity and low priority:-
  suppose we are having an application where we enter
employees record and assingn subjects they teach. Now we
have entered an new employee record and save it and assigned
subjects to it. Now if we want to modify the the subject
assigned to some other and try to save it is showing an
error message unable to save. For this scenario if we enter
the details perfectly we can complete the task and no need
to change and modification takes place rarely so we can go
through the application without any disturbance. but to
solve the problem of modication the developers has to go to
the entire code and might takes place to solve that and
might cause some side effects too. so impact is less but
solving the problem is very high . so this reason it is
called high severity and low priority.

this might help you in clarification.

Sameer Test Lead