How many lives were estimated to have been saved by the Hiroshima and Nagasaki bombs?

It's a negative number; the bombs were militarily unnecessary to secure the Japanese surrender. The US Strategic Bombing Survey, released 1946, estimated that:

Based on a detailed investigation of all the facts, and supported by the testimony of the surviving Japanese leaders involved, it is the Survey's opinion that certainly prior to 31 December 1945, and in all probability prior to 1 November 1945, Japan would have surrendered even if the atomic bombs had not been dropped, even if Russia had not entered the war, and even if no invasion had been planned or contemplated.

This was a group primarily comprised of military leaders, though with a sizable civilian contingent. In their measured opinion, the Japanese could not have sustained the war more than a few additional months.

They also present evidence that the Emperor wanted peace:

The talks by the Japanese ambassador in Moscow and with the Soviet ambassador in Tokyo did not make progress. On 20 June the Emperor, on his own initiative, called the six members of the Supreme War Direction Council to a conference and said it was necessary to have a plan to close the war at once, as well as a plan to defend the home islands. The timing of the Potsdam Conference interfered with a plan to send Prince Konoye to Moscow as a special emissary with instructions from the cabinet to negotiate for peace on terms less than unconditional surrender, but with private instructions from the Emperor to secure peace at any price. Although the Supreme War Direction Council, in its deliberations on the Potsdam Declaration, was agreed on the advisability of ending the war, three of its members, the Prime Minister, the Foreign Minister and the Navy Minister, were prepared to accept unconditional surrender, while the other three, the Army Minister, and the Chiefs of Staff of both services, favored continued resistance unless certain mitigating conditions were obtained.

The common interpretation, especially among revisionists, is that Truman and the US were aggressive in preparation for the post-war world and competition with the Soviets. I'm not expressing an opinion on this point, but I am relaying the theory.

In any case, the dropping of the bombs was needless. Japan lacked the resources and strength to keep the fight going. Blockade and targeted conventional bombing could have contained them until they finally sued for peace. It appears from the historical record that the Japanese leadership already saw the writing on the wall and was preparing for the inevitable defeat several months before the US dropped bombs on civilian population centers.

I don't think it "saved" any lives, as the bombs didn't actually end the war.

Japan was not afraid of this new technology and were not going to surrender just because a new "bigger" bomb was being used against them.

What happened was that, shortly after the bombings, the USSR declared war on Japan. Once that happened, Japan immediately surrendered because they did not have the capacity to fight BOTH the USA and USSR.

Here is an article which fully explains what happened:

http://www.foreignpolicy.com/articles/2013/05/29/the_bomb_didnt_beat_japan_nuclear_world_war_ii?page=0,0

It is impossible to say how many lives were saved by the atomic bombing of Japan because no one knows when Japan would have surrendered if the bombs had not been dropped. However, it is likely that the war would have gone on for many months and culminated in the US invasion planned for November 1, 1945. In that case, many tens of thousands, if not hundreds of thousands of Japanese would have died from the continued B29 bombings using conventional bombs. Also, remember that heavy fighting was still going on in China where it is estimated that 10,000 Chinese were dying every month. Also the Russians had just started attacking Japanese forces in Manchuria and Korea. Who knows how many casualties that campaign might have inflicted in those months. If the invasion had taken place, the number of casualties on both sides would have been very high because the Japanese were intending to defend their homeland with everything they had (including 5000 Kamikaze planes and hundreds of kamikaze motor boats. They hoped that by inflicting significant American casualties, the US would agree to peace terms more favorable to Japan. Also, the US planned to drop 3 atomic bombs as a prelude to the invasion. The effect of such an action cannot even be estimated. As a further point, if the atomic bombs had not been dropped on Hiroshima and Nagasaki, then it is very likely that both cities would have been the targets of conventional B29 bombings which probably would have resulted in close to the same number of casualties (100,000 Japanese died in the first fire bombing of Tokyo by B29s). As references for the above, I cite the following sources:

 The Fall of Japan, William Craig, The Dial Press, 1967
 Japan Subdued, The Atomic Bomb and the End of the War in the Pacific, Herbert Feis,
    Princeton University Press, 1961
 The Invasion of Japan Alternative to the Bomb, John Ray Skates, University of South
    Carolina Press,  

This is a matter of very hot debate. It depends on what assumptions you make about what would have happened in the future. But there are two basic scenarios:

1. The bombings saved somewhere in the neighborhood of 250 - 500 thousand US lives, and Japanese lives in the millions.
2. The bombings saved US lives numbered only in the thousands, and actually cost the Japanese up to a quarter of a million lives.

To my mind nobody is in a better position to make estimates of what a full-blown invasion would cost in lives than the people who were in charge of the armed forces at the time. Lukily for us, the US armed services did estimates for their own purposes before the A-bombs were dropped.

The Army estimated they would suffer 7.45 casualties/1,000 man-days and 1.78 fatalities/1,000 man-days in an invasion. So assuming the invasion of Kyūshū took 90 days, that works out to about half a million US Army wounded, and over 100,000 dead. If they then needed to invade Honshū and take the Capitol in another 90 day campaign, that would more than double the toll to about 1.2 million wounded, and over a quarter of a million US Army dead. This is about the lowest estimate you will find from Army sources, but being the most detailed I give it the most credence. But you should be aware there were numbers much higher floating around that decision-makers in the USA would have been hearing.

Interestingly, on this basis, the Army put in an order for about half a million Purple Hearts (the medal given for combat wounds of any kind). They have been working through this stockpile ever since, and still have over 100,000 of them left.

This did not cover Japanese casualties though. Presumably they would be even higher. This is all based on the assumption that the defence from the Japanese military and civilians would be simliar to what was encountered in Okinawa.

The Atomic bombings themselves are estimated to have killed about 150,000–246,000 people (almost all Japanese of course).

Both at the time and today there were those who claimed the Japanese would not have fought like they did on Okinawa, and in fact were close to surrendering in a way acceptable to the US. If they happen to be correct, then the death toll from the bombing is rather massively in the red (on the Japanese side. On the US side, it still saved lives). But we have no real way of knowing this. I know there are some documents floating around that some claim prove otherwise, but I'd posit anyone who insists some documents prove future actions of politicians has a poor (or idealistic) understanding of politicans.

Which side a person believes to me seems to have far more to do with what country that person hails from and/or how that person feels about Atomic Weaponry in general, than anything else. So the hope of there being a single impartial number that isn't somehow tainted by politics is remote in the extreme.