Bear in mind that I'm no historian, just an enthusiast that's been reading books about the American Civil War recently. I'll do my best, and I hope it's enough.
Braxton Bragg seems like an ineffective commander in highsight. Yet, as the war developed, it actually seemed like he was competent. Even when it became apparent that he was ineffective, usually something or another happened and he remained in command. We can thank Davis, who was known to value loyalty to him and hold grudges, for this. Bragg first assumed command of a 15,000 strong force in Corith. They had just been defeated under Beauregard, and were demoralized. Bragg injected some discipline into them. When Beauregard was wounded, Davis decided that Bragg was a good replacement and appointed him commander.
During the following months Bragg managed to use raids and cavalry to prevent Buell from advancing. He then invaded Kentucky with minimal losses and even inaguarated a Confederate governor. When compared with Buell, who had barely advaced at all, Bragg semed like an impressive, capable commander. Even though Bragg was ultimately forced to retreat by Buell's superior numbers, Buell suffered heavy cassualties and Eastern Tennessee was still in Confederate hands. It seemed like a Confederate victory, at least in Davis' eyes. Buell was, on the other hand, removed from command. We should also note that Bragg's invasion (or rather, extended raid) was at the same time as Lee's invasion of Maryland. Thus to many northerners it seemed like the Union was collapsing, Lee himself considering it the low-water mark of the Union.
Later, at Stone's River, Bragg sent a premature message saying he had won a great victory, which "produced great exaltation". Again, he was forced to retreat, and several commanders lost their confidence on him and wanted him removed. Davis sent Johnston there, but Johnston advised to retain Bragg, most probably because he didn't want to command an army on the west, but wanted to go and fight in Virginia. He even said that if they wanted to replace Bragg, they should send Longstreet. When the War Deparment practically demanded Johnston take command of the Army of Tennessee, he argued it would be inhumane to replace Bragg while Bragg's wife was sick, and then Johnston himself fell ill. As a result, Bragg remained as commander.
More disasters seemed to come in Bragg's way in the following months, yet no suitable replacement was found. Lee was considered, yet he was to focused on his invasion of Pennsylvania. Longstreet was finally sent there, yet Grant still took Vicksburg... and, once again, Bragg managed to escape. Ronsencrans next campaign (which even Southern newspapers called "masterful") drove Bragg all the way to Chattanooga, and then to Chickamauga. Even when Bragg's own commander didn't cooperate with him, he got lucky. You see, one of Rosencrans' subordinates didn't see that there was already a division in a part of the Union line, so Rosencrans moved a division and created a gap accidentaly. When Longstreet attacked under Bragg's orders. Rosencrans had to retreat, and though his army wasn't destroyed because Bragg was too worried about his own losses, they ended up trapped and besieged in Chattanooga. The Confederates had won a decisive battle, and it seemed like Bragg was the reason.
Bragg's commanders were furious because they thought that Bragg was incompetent, and that the only reason the Union forces were not destroyed was him. Davis himself went there and talked with the four main commanders, among them Longstreet, Forrest, and Polk. They all said that Bragg must go. Once again, there simply wasn't any suitable replacement in Davis' eyes. Davis disliked Johnston, Longstreet's choice, and Beauregard was doing a good job defending Charleston. So Bragg remained in command.
Davis finally removed Bragg from command after Grant defeated him. Grant's victory was both due to Grant's leadership abilities and Bragg's ineffectiviness. It was simply a disaster to the Confederates. Bragg himself called it "shameful" and offered his resignation to Davis, who then finally appointed Johnston, and even then, begrudgingly.
As you can see, Bragg remained in command mainly because it seemed like he was winning and because Davis refused to appoint Johnston or Beauregard, whom he despised.
Sources: James M. McPherson's "The Battle Cry of Freedom" mainly.
While I can satisfactorily answer the question I think such an assessment of Bragg is unfair. Perhaps more charitably Bragg's infamously difficult personality led to his ineffectiveness. Chiefly, Bragg remained in command because of his relationship with Jefferson Davis and the lack of a viable candidate to replace him. However, on at least three distinct occasions he offered his resignation superiors if they so desired. On one such occasion Bragg survived because Joseph E. Johnston found his army to be in stellar condition, in his learned opinion, and demurred from personally taking command on account of health issues. On the second Davis himself declined to accept Bragg's resignation despite the army's senior commanders being on the verge of mutiny. On the third, following the disastrous loss at Chattanooga, Davis simply had no choice but to relieve him of his command.
For my part Bragg is unfairly criticized for his numerous shortcomings as a commander. He was a harsh disciplinarian which rubbed a number of his subordinates the wrong way. His mood swings and irritability were at least partially a byproduct serious medical ailments that plagued him throughout his wartime service. His decision to withdraw at Perryville, while questionable, was hardly an egregious one given the information at hand. Robert E. Lee had just lost at Antietam and was withdrawing, Van Dorn and Price had just lost Corinth and were in retreat. Bragg could have gambled on an uncertain victory, but defeat posed serious dangers deep in enemy territory. Buell had only engaged with about 40% of his available manpower and reinforcements were on their way. Could he have beaten Buell's initial force in detail and withdrawn before Buell could make his numbers felt, possibly, but it was a risk Bragg believed wasn't worth the reward.
In January Bragg successfully bloodied Rosecrans at Murfreesboro, but he withdrew again. Given that he had lost a 1/3 of his men, it was the dead of winter, Rosecrans was being reinforced, and his corps commanders all counseled withdrawal why was he ridiculed for his choice. Because Bragg's subordinates all backtracked after the fact most notably Polk and Hardee who both tried to play it off like it was Bragg's choice and they had no input. This was in fact a lie and Joseph E. Johnston likely saw threw it which is why he praised Bragg's army and conduct upon his inspection.
It doesn't get any better with a host of subordinates refusing to cooperate or even communicate with Bragg leading to Rosecrans successfully maneuvering him out of Tullahoma and again at Chattanooga. As Rosecrans pursued him Thomas C. Hindman another of his generals flat ignored an order to attack an exposed wing of Rosecrans army. A few days later Polk refused to do the same unless he received more men and complained loudly about his situation. Ultimately the less influential malcontents were removed from their command including Polk, Hindman, and D.H. Hill. Longstreet, whom Bragg also began feuding with, was dispatched to Knoxville, and the complete lack of trust or cohesion on the Confederate side of things was expertly if blindly exploited by the Union to completely shatter the siege around Chattanooga. Following this disaster it was readily apparent to all parties that Bragg could no longer lead the army and he was relieved and transferred to Richmond.
This is not to absolve Bragg of any of his shortcomings, but it is unfair to suggest the nature of his incompetence or ineffectiveness was entirely one sided. Allowing your personal disputes to spill into professional misconduct on the field of battle is inexcusable. Slandering your commanding officer to his superiors by disassociating yourself from your own decisions is disgraceful. Putting petty and vindictive vengeance above the well being and success of your own men makes Hardee, Polk, Hindman, et al every bit as culpable for Bragg's failures as the man himself. Bragg was ineffective as a commander because his subordinates failed to respect him enough to follow him. That's not a one sided issue though and for all of Bragg's failures, the men who followed him are hardly blameless.