OK, to help answer this, here's a basic overview of the flat Earth theory (there is some variation on details depending who you ask):  "The Earth is a round disk with the North pole at the center and continents arranged around that with an icy brim at the edge holding the oceans in.  The sun and moons are spot lights above the disk that point at different areas during the daily cycle."  This world view fits most strongly in a fundamental religious context, particularly appealing to some Christian creationists, so many arguments are made from a religious or scripture perspective.  Publicly, flat earthers often choose to cite scientific theories, practices, and observations as evidence, but their interpretation doesn't generally agree with that of the scientific community.

* Horizon at eye level:  This may be countered with a vague statement about optics, or a claim that this isn't how geometry works.

* Ships at sea: This may be countered with a claim of angular resolution of the optics used for the observation.

* High altitude photos/videos:  Claims have been made that these aren't falsifiably/reproduceable, so, by the scientific method, they don't count as evidence.

* Lunar eclipse:  This can be described as a mystery object passing between the Earth and moon.

* The Foucault Pendulum:  Some say the Foucault pendulums are rigged with hidden mechanisms.  Some say the Foucault Pendulum motion must be due to some sort of universal motion, rather than a spherical Earth.

* The behavior of gravity: Flat earthers may say that Newton's gravity model is flawed, and only perpetuated to support the spherical earth story.  They may also claim that their concept of gravity is the same as everyone else's because we're still pulled towards the Earth and nobody is floating away.

* Time zones: The sun spotlight in the flat earth model moves during a daily cycle, so time zones are explained this way.  Alternatively, some flat earthers argue that time zones are just a man-made construct and have nothing to do with reality.

It should be noted that flat Earth beliefs are held by a fringe group of people, and not generally accepted as true.
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