Most (but not all) Jazz is played in what has become known as Swing Feel and this is something that many students initially find hard to grasp. This is often indicated at the beginning of a piece of sheet music by either the simple instruction "Swing Feel" or by this 
                                                                                                                                                                                            which means play the two quavers (eighth notes) with a triplet feel with the first one lasting for 2/3rds of the beat (in other words as a crotchet or 1/4 note) and the second one lasting for  1/3rd of the beat. Another way to think of this would be to play as though the piece was in 12/8 time rather than 4/4 and the first of each quaver is played as a crotchet and the second as a quaver, so a C Major scale written like this

                                                                             would be played  like this
Notice also that in the second example the accents have been displaced from the downbeat to the upbeat. This is a good way to practise scales but of course when playing Jazz you wouldn't literally accent every second quaver but accent some of the upbeats. Another important concept to understand is that in 4 time instead of Beats 1 and 3 being strong and 2 and 4 being weak in Swing Feel beats 2 and 4 are strong and 1 and 3 are weak. A further feature of Swing Feel is that the last note of phrases are often accented and played staccato. 
Having said all that I want to stress that the only way to fully understand Swing Feel is to LISTEN, LISTEN, LISTEN.
It is impossible to write Swing Feel exactly using conventional notation and the explanation that I have given above is at best only an approximation, for example at an average speed the triplet feel above is pretty close to Swing Feel but at ver fast tempos quavers are played closer to Straight 1/8s (in other words as written). I have put these videos from Youtube where different people explain their take on Swing Feel. The first one explains with notation pretty much as I have done and the other there place a heavy emphasis on the need to listen and also to try playing along with recordings. When trying to play along with Youtube videos remember that they can be slowed down if the piece is at a fast tempo.