The Ionian Boots of Lucidity are a favoured choice on AD casters, who prefer the CDR to the attack speed of the other main offensive AD boot choice, the Berserker's Greaves, and don't tend to get as much CDR from their other items as mages (who for this reason tend to buy Sorcerer's Shoes instead).
Many supports also pick up these boots, as their utility to their team tends to come in a large part from their abilities.
Mages can also pick those boots instead of Sorcerer's Shoes, but they have to buy other magic penetration items such as Void Staff to compensate.
When picking those boots, keep in mind that sometimes it may be better to buy other boots, such as Mercury's Treads or Ninja Tabi, as the defensive stats may be sometimes more useful than the extra cooldown reduction, or you have other ways to reach the cooldown reduction cap.
Ionian Boots of Lucidity and a fully stacked Mejai's Soulstealer are the only items that grants an amount of cooldown reduction that isn't a multiple of 10, making them work well with Sorcery. It can be used to reach the CDR cap of 40% without cooldown reduction runes or buying extraneous cooldown reduction.
The Boots of Lucidity were introduced as the item that honors the Ionian Victory in their rematch with Noxus.