Located off the Mediterranean coast of present-day Lebanon, Tyre was pretty much an ancient Phoenician Azkaban Prison. The city was an island whose walls extended directly into the water, which meant that even if Alexander had a navy with him (which he didn't), his entire army would splash as helplessly against Tyre's defenses as piss off a flagpole.
Alexander's solution to this dilemma: Simply change the map forever by making the island not be an island any more.
It sounds like something that would only work in a cartoon, since it would require them to spontaneously construct a kilometer-long land bridge to link Tyre back up with Eurasia, by hand. They did it anyway.
Slowly, and while being pelted with arrows and bombarded by Tyre's navy, Alexander's men built their new land mass, one stone at a time.
It's still there.
Once the new land mass was in place, he was able to wheel his siege towers right up to the fortress. Ships belonging to his allies eventually came to help out, possibly because they heard what they thought was a ridiculous rumor and wanted to come see if it was true.
With Tyre now checkmated, Alexander personally led the final charge against the city from the top of his tallest siege-tower. The city fell to Alexander, and with it its status as an island. You might be asking the obvious question, which is why he didn't have his men keep throwing down rocks until they'd formed a huge "ALEXANDER WAS HERE" in the Mediterranean sea--and of course the answer is that he could not have known that aerial photography would one day be invented.