Burnside had 100,000 men, Lee about 72,500.
This was a one-sided battle: Lee lost about 4,500 men, Burnside over 13,000.
McClellan had ejected Lee from Maryland, but did almost nothing with his victory. He paused to rest and reorganize (few armies can have been as well rested and tidily organized as McClellan's) and fortified Harper's Ferry. Meanwhile, JEB Stuart made another raid around the Army of the Potomac, suffering one man wounded in four days. It infuriated the northern populace, and dropped McClellan further in Lincoln's eyes.
McClellan's only apparent move was to maneuver in the Shenandoah, where he was quite successful in small actions and ' had he known it ' in splitting Lee's army, but Lincoln had seen enough. McClellan had marched around plenty of times before, and this looked like just another aimless march (and McClellan had no clear plan of campaign) so Lincoln sacked him.
Ambrose Burnside was to be the new commander (against his own wishes ' if not a great general at least he knew his own limits), and he had a few new ideas. He reorganized the Army of the Potomac, formerly seven Corps, into three Grand Divisions and a reserve, which should be easier to command. (Lee had simplified and strengthened his own chain of command by creating two Corps ' one for Longstreet, one for Jackson ' with divisions and brigades tidily arranged.) The difference was that Burnside changed most of his Corps Commanders. His new campaign plan was to move quickly eastward, sprout a new base at Aquia Creek, and cross the Rappahannock at Fredericksburg before Lee could arrive.
Once he obtained Lincoln's approval, Burnside moved quickly. He got his Right Grand Division to Fredericksburg before Lee had more than a few pickets there ' but the pontoons were late. Burnside had to wait eight days, from November 17 to 25, for pontoons to arrive. The river was fordable, and Burnside might have begun crossing his infantry, but played things cautiously. Meanwhile Lee was moving rapidly: Longstreet's entire corps was in position by the 21st. This caused Burnside to pause and concentrate more of his men ' which in turn allowed the Confederates to bring Jackson over as well. Jackson took up position on the eastern part of the line, in case Burnside tried to cross downstream of Fredericksburg.
Burnside did indeed contemplate how to feint Lee out of position, and tried some ruses to pull troops east of the town, but made little headway. Burnside also convinced himself (if not Lee) that the true surprise would come in doing the obvious: attacking at Fredericksburg. Longstreet fortified his men thoroughly, and also prepared roads behind the line so reserves could be shifted anywhere along the line. Guns were emplaced 'so that a chicken could not cross the fields'.
So on December 11 massed Union batteries on the heights north of the river opened fire to cover the engineers building bridges. Lee had felt the power of the Union guns at Antietam, and rather than put his men where they could be pummeled, had basically conceded the town and chosen a line along the line of hills south of the river. Barksdale's brigade of Mississippians, scattered through the town, were there to harass the engineers. They certainly did, causing proportionately heavy casualties and inflicting serious delay ' it took all day to build the bridges. (Meanwhile, a little downstream, Franklin's Left Grand Division crossed without trouble.) It then took the whole of the 12th for all the Union forces to cross, and by the end of the day there were about 50,000 men in the eastern bridgehead and about 30,000 in the one at Fredericksburg. The delay was plenty of time for Jackson to move his men upstream, and they extended Longstreet's right.
Longstreet's men had dug in; Jackson's men had no such fortifications, and Burnside had first intended his main blow to be on Jackson ' but at the last minute, 5:55 am on December 13th, he switched. Franklin was to expand his bridgehead, but nowhere to attack in great strength, while Sumner on the Union right was to clear the hills behind Fredericksburg ' right into the teeth of Longstreet's defenses.
The first move was on the Union left, where Franklin untangled Burnside's order and sent George Meade's division of Pennsylvanians forward. Once the morning fog lifted the whole movement was visible, and a battery of Confederate horse artillery trotted out onto Meade's flank and proceeded to disorganize everything. It took an entire infantry division to drive 'The Gallant Pelham' back, subtracting one of the two divisions intended to support Meade's charge. Franklin's batteries shelled the Confederate guns into silence, but destroyed relatively few ' Jackson had put some effort into entrenching his batteries, if not his infantry. Then Meade's charge began. He swept forward and hit a seam in Jackson's line (Jackson had not scouted thoroughly, and thought a swampy section was impassible). The first line crumbled, the reserve brigade was surprised and tumbled backward, and it took a vicious counterattack for Jackson to reform his line. Meade's open left flank provided the opportunity ' if Doubleday had done more than stare at a battery of horse artillery all day, Burnside might have had a partial victory.
Stonewall Jackson took command of the counterattack, and drove Meade back out of the woods and pursued onto the open fields along the railway tracks. He had the gleam of battle in his eyes, and he wanted to exploit the Union confusion to storm all the way to the river, cutting off and capturing large numbers. But the Union batteries were strong and ready, and inflicted about 500 casualties. Fresh Union divisions replaced the battered front line, and Jackson's men fell back into the woods.
Meanwhile around Fredericksburg a separate battle was being fought and won ' entirely by the Confederates. When the fog burned off the Confederates on Marye's Heights had a grandstand view of the Union troops clogging the streets of the town below. And the Confederates up there were artillerymen. They began firing into the town, completing the wreckage after the Union bombardment of the 11th and plundering on the 12th. Sumner's men pressed through the streets in column, and had to stay in column to cross a narrow but deep mill-race that carried water from an upstream dam to power waterwheels at a small factory. They crossed under shellfire on the two bridges available, then had to turn right along the stream to deploy under shellfire, then they charged towards Marye's Heights. But at the bottom of the hill was a small road that had sunk a few feet below the original ground level; by the side of this road was a low stone wall. The combination was enough to give excellent protection to the Confederate brigade (soon reinforced) deployed in the road. Successive Union brigades charged the road, and they got successively closer, but not a single man managed to make it to the stone wall. The Confederates were eventually four, even eight, deep, so there was a constant heavy fire as some men loaded and others fired.
Casualties were terrible; Hancock's men left 2,013 of their 5,006 on the battlefield. And Burnside didn't realize what was going on. In midafternoon he wanted Franklin to make another, much larger attack (or he may have; yet again Burnside's orders are imprecise). But Franklin got the news too late in the short winter day to do anything.
Lee was always looking for a way to turn the victory into a crushing blow, but Jackson's abortive morning attack had showed the strength of the Army of the Potomac. Jackson came up with the idea of a night attack (asked how he would identify friend from foe Jackson replied that the Confederates could attack naked) but Lee vetoed it.
The battle was over. Burnside's losses were over 10,000 against Confederate losses of about 5,000. Well over half the Union losses were in front of the fatal stone wall. When he realized the enormity of the slaughter, Burnside was stunned. At first he thought of continuing the attack the next day, leading it personally to clear his own honor. But his subordinates talked him out of the nonsense. One more casualty would have hardly made much difference, but another charge would have cost hundreds.
Burnside withdrew his men across the river on the night of the 15th, leaving only graves behind. He'd lost a battle, and with it a good portion of the confidence of his men. Yet Lee had won little. He still held the shell Fredericksburg, his men (and Southern civilians) had another victory under their belt. But there was no collapse in Northern morale. It was an attritional victory, and the south needed more than that.