good candidates for automation

Tests that should be automated:
Business critical paths - the features or user flows that if they fail, 
cause a considerable damage to the business.
Tests that need to be run against every build/release of the application, 
such as smoke test, sanity test and regression test.
Tests that need to run against multiple configurations — 
different OS & Browser combinations.
Tests that execute the same workflow but use different data for 
its inputs for each test run e.g. data-driven.
Tests that involve inputting large volumes of data, such as filling up 
very long forms.
Tests that can be used for performance testing, 
like stress and load tests.
Tests that take a long time to perform and may need to be run 
during breaks or overnight.
Tests during which images must be captured to prove that the application 
behaved as expected, or to check that a multitude of web pages 
looks the same on multiple browsers.
Generally speaking, the more repetitive the test run, the better 
it is for automation.

Also some tests are not the only candidates for automation. 
Tasks such as setting up or creating test data for manual exploratory 
testing are also great candidates for automation.

Tests that should not be automated:

Tests that you will only run only once. The only exception 
to this rule is that if you want to execute a test with a very large 
set of data, even if it’s only once, then it makes sense to automate it.
User experience tests for usability (tests that require a user to 
respond as to how easy the app is to use).
Tests that need to be run ASAP. Usually, a new feature which is 
developed requires a quick feedback so testing it manually at first
Tests that require ad hoc/random testing based on domain 
knowledge/expertise - Exploratory Testing.
Intermittent tests. Tests without predictable results cause more 
noise that value. To get the best value out of automation the tests 
must produce predictable and reliable results in order to produce pass 
and fail conditions.
Tests that require visual confirmation, however, we can capture page 
images during automated testing and then have a manual check of the images.
Test that cannot be 100% automated should not be automated at all, 
unless doing so will save a considerable amount of time.