Basic rules are pretty basic. Boots are all about either amplifying your build or role OR counterbuilding the enemy composition.
:BOOTS WITH BOOTS:
[More Movement, Please]
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Boots of Mobility are the go-to for most junglers, supports, and roaming assassins that need to get around the dead zones of the map quicker. Remember that movement is gutted to Speed 1 boots
during combat.
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Boots of Swiftness are nice for the heightened movement speed and slow reduction (ie against ZOC slow comps that may include Ashe, Kog'Maw, Lulu, Nunu, etc) OR for overriding innate self-slows (ie Tahm, Vi, Varus). These are also good for champions that want Boots of Mobility but don't want to sacrifice in-fight movement bonuses.
:DEFENSIVE BOOTS:
[Fighters, Tanks, or just plain Counter Building]
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As good as Mercury Treads are, I don't really build them anymore because of Elixir of Iron and masteries and go for a stat that I may need more. However, these give that extra MR and Tenacity to those who won't go with the tankier route.
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Ninja Tabi are excellent in a lot of situations since almost every team will have an ADC that will Auto-Attack.
:OFFENSIVE BOOTS:
[Give me that dps!]
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Berserker's Greaves are almost required on most traditional ADCs. (Caster-carries like Ezreal, Urgot, and even Lucian can build Lucidity and be just fine, though.)
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Sorcerer's Boots are for damage dealers that either thrive on base penetration (ie Rumble or Elise) or already have a 20% CDR item (Athene's and Morello). This can also be really good to snowball a magic-damage lane if you don't quite have enough for that NLR but need the mobility instead of brute damage.
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Boots of Lucidity are fantastic when your core build doesn't include innate CDR or just needs that extra top-off (like hardcore damage builds that don't go for a 20% CDR item).