Invoked as an interactive login shell, or with --login
When Bash is invoked as an interactive login shell, or as a non-interactive shell with the --login option, it first reads and executes commands from the file /etc/profile, if that file exists. After reading that file, it looks for ~/.bash_profile, ~/.bash_login, and ~/.profile, in that order, and reads and executes commands from the first one that exists and is readable. The --noprofile option may be used when the shell is started to inhibit this behavior.
When a login shell exits, Bash reads and executes commands from the file ~/.bash_logout, if it exists.
Invoked as an interactive non-login shell
When an interactive shell that is not a login shell is started, Bash reads and executes commands from ~/.bashrc, if that file exists. This may be inhibited by using the --norc option. The --rcfile file option will force Bash to read and execute commands from file instead of ~/.bashrc.
So, typically, your ~/.bash_profile contains the line
if [ -f ~/.bashrc ]; then . ~/.bashrc; fi
after (or before) any login-specific initializations.
Invoked non-interactively
When Bash is started non-interactively, to run a shell script, for example, it looks for the variable BASH_ENV in the environment, expands its value if it appears there, and uses the expanded value as the name of a file to read and execute. Bash behaves as if the following command were executed:
if [ -n "$BASH_ENV" ]; then . "$BASH_ENV"; fi
but the value of the PATH variable is not used to search for the filename.
As noted above, if a non-interactive shell is invoked with the --login option, Bash attempts to read and execute commands from the login shell startup files.