Portal Beta refers to Portal during its development stages from mid 2005 to its release on 9th October 2007. After all the puzzles were finished, the art style went through a lot of changes from one similar to Half-Life 2's Nova Prospect, to the now known Aperture Science clean look. Betas were playable and most of the chambers didn't have bugs.
Name depends on the era of the development. For example: Portal 2005 beta, Portal 2006 Beta etc., however the most common use for Portal Beta is Portal-Project Beta which is a mod that was developed by ReepBlue which contains all maps from each era.
Puzzles were first finished during Portal Development, so the beta was similar to the Retail. There are many versions of test chambers with same puzzles however the plot was different.
Player would wake up in the relaxation vault and would wait 2 minutes until the ticking analog clock finishes. Player would solve all test chambers as in retail and then escaped to the Nova Prospekt facility (lab_lvl maps). During lab_lvl maps, player would approach test chamber 16 with turrets (During that time of the development, Test Chamber 16 wasn't even a test chamber but it was rather an escape map). After the turret chamber player would approach the hallway (lab_lvl3a) and was supposed to find 3 axillary units and put them in the axillary unit chamber to unlock the door which later leads to test chamber 18 (Which was an escape map by title lab_lvl4 and had the same layout as retail). The ending of 2005 is unknown or wasn't even finished.The best mod that recreates this era is Portal: Prospekt.
Early 2006 (Leipzig)
After the developers rejected the Nova Prospekt design, things were slowly changing to the clearer look. Leipzig era kept the combine elevator and the analog clock was still there. The doors took a shape of Star-Trek looking doors. Chamber 13 is the only map that is available from this build.
Portal in 2006 looked very similar to the final product. GLaDOS' voice was different and in some chambers more distorted. Another difference between 2006 and the 2007 Final retail is the escape: The beginning of the escape is the exact same as in the Retail with few changes within the details. The player would later encounter GLaDOS which took an appearance of a rollermine cube and she would talk about the Black Forest cake recipe (that dialogue was later recycled for the blue personality core in the 2007 Retail Escape). The player would redirect the rockets from the rocket turret (like in the retail) to GLaDOS and she would escape to another room where obstacles are more serious. In the second room which is smaller, turrets would still appear and the rocket turret was on the ceiling. When GLaDOS is defeated, the room gets flooded and the room transforms into a dirtier and less finished room and the water would go away. The player could get out. The reason that ending was cut was because play-testers felt guilty chasing something that would later be helpless and killed.
Very close to the final product with lighting changes and the final core where the player confronted GLaDOS was looking less finished (notice the incinerator).