In summary? Healing surges denominate the amount of "spare life force" that a PC or NPC has available at the start of each day. As most healing powers work along the lines of "spend a healing surge and regain healing surge value + x hit points", this affects how often a character can heal themselves each day and gives incentive to not be too careless with their health.
Healing surge value, incidentally, is rulespeak for "one quarter of your maximum hit points (rounding down)".
PCs have healing surges to spend equal to their class level + their Constitution modifier - squishy classes like the Wizard or the Warlock tend to get fewer healing surges, as per the tradition, but the difference isn't as major as you might think. For example, in the first PHB, the Defenders (Paladin and Fighter) have 10 and 9 healing surges, respectively, whilst the Leaders (Warlord and Cleric) have 7 healing surges and the Wizard and Strikers (Ranger, Rogue, Warlock) have only 6 each.
Like many 4E mechanics, this was prototyped in other Wizards of the Coast products, in this case Star Wars: The Roleplaying Game Saga Edition's "Second Wind" action. Also like many 4E mechanics the prototype was better received than the actual thing. Key differences include that Second Wind is far more limited (1 day unless you take a feat to get a second), and they are the only source of healing besides medical treatment and zero-sum force powers in the game, whereas in 4e, there are some methods of healing that don't need a healing surge to work.