Ron Miller, as reported on the Planetary Society Blog, made some "photos" of places on earth if earth had saturn's rings. Mr. Miller is no neophyte to such visualizations; he is good at what he does. In any case, people are going to notice those rings even if they're much, much smaller. This question really has a vast number of good answers, but take this answer as a short list of very major effects.
The Cultural Significance
From my brief survey, stars culturally affect our lives in many ways; from nursery rhymes to astrology. The Sun and Moon were often gods in various cultures, and therefore are quite important. We used the sun to mark time before the industrial revolution, and some cultures historically use lunar calendars. It ought to be pretty obvious from any amount of research into the cultural history of the Sun, Moon, and stars that they play a huge role in cultures the world over. Adding in some rings to an earth-like planet could include the following cultural effects:
- A New God (in cultures that think heavenly bodies are gods...). Yes, ancient gods play a very diminished role in our current society, but in ancient societies they were quite a big deal. That's like having another major religion or political party today! Obviously such a thing so early on would have huge impacts on culture.
- New Religious rites; each religion in the world can react or think of the planetary ring differently. Some religions have holidays or events due to celestial events or timings. The ring could alter people's concept of heaven or other afterlives, too!
- A new item by which to tell time. That's more of a science thing, though, but it impacts culture quite a bit!
- New Iconography. The rings of a planet could symbolize something, much like how some people think of the moon as representing sleep, peace, etc. and the sun representing truth, light, or power. Maybe the rings could represent consistency or endurance? It's hard to say how people would use this in their poems, songs, art, and religions. What does that ring mean to these people?
- Architecture may want to conform or use the rings to increase particular effects. Using the ever-present rings may significantly change how people build their buildings. For instance, the golden ratio may not have been as important to the greeks. Perhaps they valued the ratio or shape of the rings from their position; therefore old, authoritative, and/or venerable institutions would try mimic the ring instead of having this ratio! This simple change could affect how streets, cities, and buildings are shaped and laid out.
- Faster Exploration of the Globe. If the ring appears to touch the ground, you may want to find where the ring actually touches. Since a ring never touches its planet, but people are looking for where it does this impossible thing, they will travel more and further. Imagine if the Columbian Exchange happened earlier, or if the European diseases never killed off such large portions of Native Americans because they had already been exposed to them before Columbus! (The colonial age could have never happened!)
It could be useful, for at least this answer, to think of science as figuring out what is going on in the world. Humans appear to have a basic need for some level of understanding for what is going on in their world. This figuring out has lead to astronomy, chemistry, physics, and the world we have today. Introducing a new element, such as the rings, gives people a new thing to examine and ponder. Here are some things to consider:
- Navigation may be easier. Since the ring(s) won't change their position much, you can use the rings to guide yourself. Assuming the rings go east-west and are roughly at the equator, they will appear in the southern sky in the northern hemisphere. This makes finding your way much easier! Also, some quick math and measuring the angle of the ring will let you determine your latitude.
- The concept of heliocentrism may been harder to accept. After all, if we're not the center of the universe, why do we have a ring around us? With such a simple defense, it may be quite a while before people take heliocentric models seriously. It should be noted that heliocentrism (as proposed by Copernicus) was made in 1543, but wasn't proved until 1723 by James Bradley. Bradley gained hard proof by using the parallax method to measure star distance. The parallax method relies on the heliocentric model, and its results cannot be reasonably duplicated by a geocentric model. Shadows on the ring only prove that the sun is a major source of light; it says nothing about what is revolving around what!
- The Flat Earth Model may never really have taken off. Especially once people realize the rings never touch the earth but are so obviously curved!
- Another system to tell time. Depending on the regularity of celestial events, the ring could be used to tell time better than the sun or moon. I suspect such time-keeping would likely be just as accurate as solar or lunar calendars, because the ring's behaviors are likely not dependent upon the sun/moon cycles, and may just appear to be a static feature of the sky.
- Less dark nights could happen; the ring could reflect more sunlight (more regularly) than the moon, allowing creatures to evolve less powerful night-vision. After all, if there is ample light, what evolutionary pressure is there to see in the dark? Maybe color would become more important to the success of more species!
You should note that I have highlighted older ideas and focused on things which occurred in the far past. This is simply because the presence of this celestial object would impact so much of human life so early that it would be quite impractical to figure out what an earth-like planet with human-like people would look like beyond a certain limit. It simply can affect so much that no answer can include all the effects!