In a 4/4 situation, how does "importance" work in beats? I read somewhere that rock puts emphasis on certain beats and jazz puts emphasis on other ones.
Also, when we have an important beat, how does it influence the music? Would you more likely have a melody whose note is a part of the chord or something?
Well, they are all important, but there is a non-genre specific natural human tendency to understand rhythms as duplex, meaning divided in halves repeatedly, and to consider the first half of any subdivision as different from the second half, usually with an emphasis on the first half.
As an example, let us consider a measure of music. As we all know, it's exceedingly common to divide into two parts and then divide those parts into two parts each, giving us four total division of equal amounts of time which we call 4/4 time, or common time (furthering the notion that this is intuitive and popular).
Those two layers of division lead to two layers of beat importance (to coin a phrase), like so:
ONE two three four
The one naturally has the highest importance, the three slightly less, and the two and four have about the same, lesser importance. Further subdividing into eighth and sixteenth notes creates more layers of importance.
My meaning of the words "naturally" and "importance" is meant to speak of an intuitive preference or emphasis. To understand why this might be a human convention, consider walking. We have two legs, and when we walk, we can notice that either our legs slightly differ, or our use of each leg slightly differed, or both. We naturally create our own walking rhythms that are not 100% symmetrical between our two legs. So it is understandable that humans would be prone to dividing things into two, not exactly equal parts.
Different styles of music do play with these concepts. Placing the emphasis on non-intuitive time divisions (syncopation) or using non-duple divisions (tuplets and compound time) are common methods used to make music more challenging and exciting. Taken to extremes, music can be made to feel esoteric and inaccessible (to many, if not all), while following the our intuition closely yields a more basic, pop feel.
Regardless of the rhythm, placing notes on the more important beats highlights those notes. Really I should say that placing the start of a note on a more important beat highlights that note. That also applies to placing non-note information on or off the beats, such as lyric syllables and percussion hits. The effect is subtle but palpable.
To that last point, if you give this video about five minutes of your time, you'll see/hear/feel what kind of difference an understanding of our instinctive sense of emphasis on words, beats, and notes can make:
