When the Catwoman first appeared in Batman #1 she was simply a glamours burglar known as the CAT.

For her second appearance in Batman 3 she donned a fearsome Cat costume and became the Catwoman!

from "The Batman vs the Catwoman" by Bill Finger, Bob Kane , Jerry Robinson and George Roussos Batman 3, Fall 1940

She continued to wear this outfit, although the color scheme varied, throughout the war years.  In Batman 35, she adopted the look which would remain for the rest of the Golden Age.

from "Nine Lives has the Catwoman by Bill Finger, Bob Kane and Ray Burnley. Batman 35, June 1946.

The Catwoman's last Golden Age appearance was in September 1954 in "The Jungle Cat Queen" in Detective 211.
The first Silver Age Catwoman appearance, surprisingly was in a Lois Lane  comic, released at the height of the Batman TV show boom, when every one was trying to get into the act.  

The traditional outfit was preserved with the addition of cat's claw gloves.

from "The Catwoman's Black Magic" by Leo Dorfman and Kurt Schaffenberger.  November 1966

Catwoman finally surfaced in Batman 197, in December 1967 in an outfit based on the Julie Newmar character from the TV show.

from "Catwoman Sets Her Claws for Batman" in Batman 197 by Gardner Fox, Frank Springer and Sid Greene.

The new outfit didn't last long and was soon replaced by this one
from "The Case of the Purr-loined Pearl" Batman 210 March 1969 by Frank Robbins, Irv Novick and Joe Giella.

By Batman 291 in September 1977, Catwoman had returned to the purple and green outfit of the late forties and fifties which would remain her most classic costume.  She continued to wear this outfit until 1986 when she dropped the cape.