There isn't enough information in this question to give an answer. The only certainty is that a local or remote connection doesn't really affect how authentication is performed.
Short Answer
Make sure you are connecting to the correct server. If you are, check SQL Server's log for the failed login message. The state number of the message explains the failure reason. In the SSMS Object Explorer, go to Management > SQL Server Logs > Current.
Long Answer
The fact that the error is 18456 means that you can connect to a server but the credentials are rejected.
One reason for this may be that you are connecting to the wrong server: when connecting to an instance, you must always provide the instance name, eg: .\INSTANCE or MACHINE\INSTANCE. If you click on the Technical Details
button in the SSMS error dialog, you'll see what server\instance you tried to connect to. For security reasons, no other information is disclosed in this message.
BTW, clicking on the help button of the dialog box will open a help page with a quick explanation of what might have happened.
To diagnose a real authentication problem, you need to find the full error message. The full message, including a state number that explains the reason, is stored in the server's Log. In SSMS, you'll find the log in Management > SQL Server Logs > Current.
In general, checking SQL Server's log should be the first action when an error is encountered that isn't directly related to a SQL statement.
The meaning of the various states is discussed in this similar SO question, described in this MSDN blog post and this detailed article by Aaron Bertrand.
A quick list of state numbers, copied from the SO question:
2, 5 = Invalid userid
6 = Attempt to use a Windows login name with SQL Authentication
7 = Login disabled and password mismatch
8 = Password mismatch
9 = Invalid password
11, 12 = Valid login but server access failure
13 = SQL Server service paused
18 = Change password required