What would a planet's rings look like from the surface of that planet?
Would the rings be perceived to be stationary or move across the sky, and would the alignment of the rings relative to the axis of rotation affect this?
Very cool, I'm sure. Roy Prol put together a fantastic animation describing what a Saturn-like ring system would look like from the surface of the Earth:
Essentially, the rings would most likely be aligned with the planet's equator running from east to west. When viewing the rings directly from the equator, you would be seeing them perfectly edge on, appearing as an impossibly thin, bright white line running directly overhead from horizon to horizon.
But as you move farther from the equator, the rings would appear wider and wider and closer to the horizon. In the northern and southern latitudes, you would be able to make out the individual ring patterns, but the rings themselves would not appear to move to the naked eye. You're looking at billions of particles ranging from maybe a few meters across down to micrometers in size viewed from thousands of kilometers away.
The surface of the rings would look like a translucent band of light made up of a pattern of concentric rings vaguely similar to the rings you see on a record album. It would put on a fantastic light show as the sun or moon moved behind the rings shimmering in varying intensities of light as the density changes from ring to ring, all while we enjoy seasonal periods in its shadow.
In an Earth-like system, the pattern most likely would not be quite as complex as those seen on Saturn because we have fewer natural satellites (one moon) which causes many of those patterns. But you would likely see a few "ripples" in the ring system as the gravitational resonances of the moon and other objects in the solar system would seem to bend small regions of the rings into wave-like patterns.
As you approach the twilight hours around sunrise and sunset, a portion of the brightly lit band would abruptly break off into pale shadow as part of the arc passes out of the sunlight into the the shadow of the planet. The early night and pre-morning hours would be brighter than what we have now as portions of the brightly lit ring would remain in the sky long after sunset, and appear again well before sunrise.
But throughout the night you would always be able to see the rings, as they would be illuminated by the moon, internal reflections and atmospheric effects reflected from the planet.
Of course, this all assumes the planet is rotating in roughly the same plane as the solar system. Planets can be tidally locked and not rotate at all (relative to the sun). And if we were talking about a planet with its axis tilted nearly perpendicular to its sun (like Uranus), you could end up with a situation where the rings were constantly illuminated, and the normal transition of "day and night" might not apply.
I had one major addition, related to seasons and tilt, that seems to have been (partially) overlooked:
The rings (assuming equatorially bound rings and a planet with an axial tilt) can shadow themselves. They have a 'top' and 'bottom' (or north and south if that makes more sense to you), and the star(sun) will only shine directly on one side at a time, and the intensity will change, depending on the season.
So an observer in the northern hemisphere, during a summer day, would see the complete ring, at its most reflective (but due to daylight washing it out, it would not seem to be the "brightest"), that same observer, late that night, would see the (apparent) brightest rings of the year, but with the shadow of the planet falling on them, so they would not appear 'complete' but would have the dark gap mentioned by the answer selected by the OP.
The same observer, in the same location, on a winter day, would not see the rings themselves, directly, but might see only their silhouette (like a ring eclipse), or possibly a strip of sky darker than the rest, depending on if they were directly in the shadow that the ring, or on the edge of the shadow, or not near the shadow. And at night in winter, they would see nothing of the ring at all, other than the black strip of missing stars.
From the spring equinox, the rings would slowly brighten until the summer solstice, and then start to fade again until the autumn equinox when their brightness is reduced to its lowest point, where it will remain until spring.