Answering how widely used they were for US units, what you're looking for is a Table of Organization & Equipment (TO&E or TOE), which describes how a unit should be organized and equipped. Here is an example TO&E for a US armored infantry company in September 1943.
The bazooka is identified as "Launcher, rocket, 2.36-inch, M9 " which is at line 99. This specifies that each rifle company is supposed to have a total of five bazooka teams assigned to it (two men in a team) - one for each rifle platoon and one each for the mortar team and LMG team. In addition, there were three more bazookas attached to the company HQ, for a total of 18 in the company.
Keep in mind that this was the ideal organization - new units were ideally to be equipped in that fashion but units actively in combat might not be re-equipped until much later, depending on the supply situation and how available the weapons were (shortages of weapons might mean even new units would use older versions until enough new ones could be produced). German army units became increasingly disparate from their TO&Es later in the war due to the worsening supply and manpower situation.
I think the closest thing to what your asking would be HEAT (High-Explosive Anti-tank ) weapons , such as the German Panzerschreck. http://upload.wikimedia.org/wikipedia/commons/c/c9/Bundesarchiv_Bild_101I-671-7483-29%2C_Reichsgebiet%2C_Soldat_mit_Panzerabwehrwaffe.jpg
Which some infantry units would be equipped with. The Americans also used the "stovepipe" or the M1A1 Bazooka , these weapons were generally used in places of high armoured combat, such as the eastern front (The Soviets would largely be using American HEAT weapons), or North Africa. How effective some of these weapons were are debatable as a US general visiting Tunisia in 1943 asked his the men on the front the same question and they replied they had not recalled seeing them stop a tank. However, although these weapons were not very useful against tanks ,against fortified positions such as a German machine gun nest , they were effective. They could be also effective if used against lightly armoured vehicles or were used against lightly armoured spots on tanks such as the underside , top or back.
The effectiveness of most infantry based anti-tank platforms was varied, and generally ran along the lines of, "the alternative was having nothing at all."
Bazookas are classified as a type of Recoilless Rifle. Recoilless Rifles had their start just before WW1 (http://en.wikipedia.org/wiki/Davis_gun), but by WW2 (if you lump the Winter War in, then the Soviets did use it first, and some capture by Finland would end up in Germany) had been established as an ideal platform for anti-tank ordinance. Some of the first used in the second world war were actually made by the Germans- they had a 7.5cm Recoilless Gun (http://en.wikipedia.org/wiki/7.5_cm_Leichtgesch%C3%BCtz_40 http://en.wikipedia.org/wiki/10.5_cm_Leichtgesch%C3%BCtz_42)- when they invaded Crete, which was deemed so effective they made a 10.5cm variant as well. The US army would then base it's Bazooka on those (loosely, mind you) which would then be re-adopted by the Germans as the Panzershrek.
The only thing that properly made the Bazooka and the Panzershrek unique were their use of rocket propelled AT ordinance.
One of the most unspoken things about infantry based AT ordinance in WW2 was the fact that generally everyone was lacking for a portable, versatile platform. Early in the war- and even later as they'd remain effective if you knew where to aim- high caliber AT rifles were common. The British, Germans, and Soviets all used them extensively. Outside of that there were a lot of unusual ideas for how to deal with tanks though, and very little established training.
(all terms used from here out can be found with simple Google searches if you want to see more.)
The British had the idea of using sticky bombs- essentially a glass vial of nitro-glycerine with a detonating devise that would be coated in an adhesive, and surrounded by a thin sheet metal cover which would be pulled off when it was to be used. It was hard to use, and had dubious effectiveness. One of it's biggest problems was that dirt on tanks would make it difficult to stick. Tanks got dirty.
The British also had the PIAT by '43. It was based, loosely, on a spigot mortar, and used a combination of a spring loading and propellant powered rounds to deliver high explosive anti-tank munitions to it's target. While it had the advantage of not leaving a smoke trail, and being relatively cheap to produce, it also had the issue of being extremely difficult to load, and it had a reputation for bruising the user. It also had some questionable capability to actually penetrate armor. None the less it stayed in service till the 50's.
When the British first entered the war they were also training home defense reserves to use molotov catapults. Unfortunately because of this business in Ukraine I'm lacking for an easily accessible picture of one that's relevant to this discussion. It was about as practical as you can imagine.
The Soviets used a mixture of improvised weapons (the molotov never really lost it's effectiveness at disabling tanks), thrown ordinance in the form of AT grenades, and rifles. While they got AT ordinance in the form of PIATs and Bazookas, they weren't well received. Soviet observers recounted that there was scant little training performed in infantry divisions for the US and British for actually dealing with tanks, and often elected to instruct (never demonstrate) them to improvise. So yes, we know that even the Soviets were aware that the US was telling it's troops to dump some TNT in a sock, coat it in axle grease, and try and use it to blow treads off tanks, as per Saving Private Bryan.
Pretty much everyone except the US had some form of anti-tank rifle. Prior to Barbarossa or so, most tanks simply didn't have armor thicker than 30mm or 40mm. A high caliber AT rifle- such as the German Panzerbuchse 39- could punch through 25mm of armor at 330 yards (which is about the same effective range as that PIAT I mention above.) The British used Boys Anti-tank Rifles. The Soviets had the PRTS and PRTD series rifles. The PTRS could punch through 40mm of armor at 100 meters, if you could get that close.
Aside from the Panzerbuchse and those Leichtgesch mentioned above, the Germans also developed Panzerfaust and Panzershrek for dealing with tanks, along with a wide array of mines that I won't get into. They also had a goofy magnetic grenade referred to as a "safe cracker" but I won't go there. Panzershreks were borrowed and reverse engineered from the US Bazooka and employed a larger 88mm round. The Panzerfaust was an inexpensive single-use warhead. While people like to quote the "6% of British tanks at Normandy" figure to discredit the Panzerfaust, as many as 70% of Soviet tank losses in the east in urban combat were lost to Panzershrek and panzerfaust.
And then there was the US, who used the Bazooka. Aside from a critical flaw in early versions where dud rounds were somewhat common, along with some sticking issues where the rocket would stick in the tube, and detonate prematurely, it was about as reliable as anything else. While it had favorable penetration figures, with the earliest versions of the 60mm rocket being able to punch through around 70mm (the M9 introduced later in the war could deal with 100mm) of armor at most ranges, it had the issue of being wildly inaccurate beyond around 140 meters. Patton told his troops to avoid using it further than 40 yards. Anecdotal reports from the field tended to follow the line of thought that the Panzershrek was better in every way.
And that highlights both what portable infantry based AT weapons were for, and why the terms for what quantifies "effectiveness" needs to be redrawn. A bazooka wasn't being used to hunt tanks, but instead make sure that tanks didn't completely overwhelm infantry, and then flank high-value targets like other tanks, field guns, ect, and often they weren't used to neutralize a target so much as disable it. In that sense the bazooka was a good weapon. If you wanted something that could take out a tank from 1500 meter, you may want a larger platform though. Eisenhower didn't call it one of the weapon platforms that won the war because it was a bad idea.
Others have already answered how many troops received the Bazooka, so I wont go there.
