To understand the Darlington pair, firstly, we must understand that one of the important properties of a transistor, is GAIN (often written as hFE). This is the ratio of Collector current to Base current.
Taking your TIP35A as an example - it has a nominal gain of 40. That is, for every milliamp flowing into the base, 40 milliamps can flow into the collector.
A Darlington Pair, is 2 transistors wired, such that their gains are effectively MULTIPLIED together. You would normally have a smallish transistor, and a larger transistor. Now, in an NPN darlington (as we are dealing with), you feed base current into the smaller transistor. The collectors are wired together. The emitter of the small transistor is connected to the base of the larger transistor. Thus, if you put them both in a black box, connected as described, you would have something that has one base, one collector, and one emitter. In other words, two transistors, ACTING like one transistor, but with much larger gain.
The whole point of doing this, is now, instead of needing to feed, say, 1A into the base of your large transistor, you now only need to feed the pair, say 10mA (these numbers are just examples), which is within the capability of your 555.