No, it's not true, at least in Europe. I've written about this before a couple of times, the most relevant thread is probably here. To quote from that:

But we can be pretty sure that a spherical Earth was commonly known from a couple of sources:

• the Elucidarium written by Honorius Augustodunensis around 1120 refers to the Earth as a sphere. The Elucidarium was a manual for the instruction of clergy, and was translated into multiple languages. (I'm copy-pasting these from Wiki because I don't feel like retyping them: "Middle English, Old French, Middle High German, Old Russian, Middle Dutch, Old Norse, Icelandic, Spanish, and several Italian dialects.") The lesser clergy were those most likely to preach and minister directly to commoners, of course.

• We have surviving sermons from Bertold von Regensburg, a German cleric, that refer to the Earth as spherical (not in the sense that he's revealing a great mystery of the Earth, but as an analogy that his hearers would be familiar with). His sermons were in the vernacular German dialect of his time period (mid-13th century), so there's evidence that he was using that metaphor in the local language that the local people would be aware of.

It's really hard to figure out what "most people" thought in eras where "most people" weren't literate. Most evidence we have (as referenced in /u/jschooltiger 's post above) indicates that even common people would have understood the world as being spherical. That being said, any literate, educated person would definitely have known that the Earth was round. They would have understood cosmology in roughly the same way the Ancient Greeks (whom they would have studied) - a spherical earth in the centre, with concentric rings of things surrounding it, the outermost rings being the "fixed stars."

In terms of geography, they understood the world as represented (roughly) in the T and O maps. Please note in most of these maps East is positioned at the top. There were three continents - Africa, Asia, and Europe, separated by a river (usually the Nile) and the Mediterranean. The T and O map were supposed to be a schematic rather than geographically useful (by and large - though some included much more detail). They represented both religious and physical topography (placing the Garden of Eden at the "top" in the far East). The rest of the planet was imagined as being covered by a massive ocean. There is also a literary tradition of an Antipode roughly on the opposite side of the Earth from Europe, which is sometimes referenced in travel literature. T and O maps represented a very powerful understanding of the Earth that was slowly devolving in the 15th century, as more and more geographically accurate information become available via more regular travel to the far East.

There were debates in Columbus's time about the size of the Earth. Eratosthenes supposedly calculated the size of the Earth quite accurately (there is debate whether is method was accurate or just lucky). Colombus is believed to have had access to this measurement, but believed it to be incorrect. The consensus was that the planet was about the size that it actually is. Columbus believed it was about 1/3 smaller, which would make a sea voyage across the other side of the planet possible. EDIT: Please see /u/jschooltiger's excellent comment below for more information. Columbus believed the earth was smaller, was wrong, but luckily for him there was a previously-unknown continent in the way that he ran in to. I'm getting to the edge of my area of expertise on this, so hopefully someone with a better knowledge of early modern geographical knowledge can flush out the last sections.