From man ps, the manual page of the ps command (excerpts only):
a Lift the BSD-style "only yourself" restriction, which is imposed
upon the set of all processes when some BSD-style (without "-")
options are used or when the ps personality setting is BSD-like.
The set of processes selected in this manner is in addition to
the set of processes selected by other means. An alternate
description is that this option causes ps to list all processes
with a terminal (tty), or to list all processes when used
together with the x option.
x Lift the BSD-style "must have a tty" restriction, which is
imposed upon the set of all processes when some BSD-style
(without "-") options are used or when the ps personality
setting is BSD-like. The set of processes selected in this
manner is in addition to the set of processes selected by other
means. An alternate description is that this option causes ps
to list all processes owned by you (same EUID as ps), or to list
all processes when used together with the a option.
u Display user-oriented format.
So the a argument lets ps show processes of all users instead of only the current user, if the processes are connected to a terminal.
The x makes ps also include processes which are not connected to any terminal in the list. So the ax together causes ps to list all processes without restriction.
The u simply changes the output formatting and visible columns.
As @steeldriver mentioned correctly in his comment, ps is a bit special because it supports both arguments in BSD style (a) and GNU style (-a). Therefore ps aux is not exactly the same as ps -aux, although it may be implemented to do the same for easier migration. The relevant paragraph of the man page says:
Note that "ps -aux" is distinct from "ps aux". The POSIX and UNIX
standards require that "ps -aux" print all processes owned by a user
named "x", as well as printing all processes that would be selected by
the -a option. If the user named "x" does not exist, this ps may
interpret the command as "ps aux" instead and print a warning. This
behavior is intended to aid in transitioning old scripts and habits.
It is fragile, subject to change, and thus should not be relied upon.
man ps, but I believea== all processes,x== all processes own by user but witha== all processes for allu== all user in userlist. – George Udosen May 01 '17 at 22:53