If I've made any mistakes below I hope someone more knowledgeable will correct me, down-vote me or whatever. :)
First check out these two sources for some nice graphs of voltage and current for caps and inductors.
Now for a capacitor you charge it up by applying a voltage source in this case your sine wave. Well the rate that a capacitor charges is directly related to the rate of change of the voltage source you apply (see sources below). When you begin to apply your sine wave at T(0) you are at the maximum rate of change and and thus the capacitor is storing charge at the maximum rate (for this applied sine wave anyway). So here you are charging up as fast as you can which means current is screaming into the capacitor. In truth electrons are flowing into one plate and out of the other all while creating an electric field between the two plates. So current flows but it does not flow between physically through the space or dialectric between both plates.
Now as you go along your sine wave the rate of voltage change is decreasing T(1) in diagram so less current is flowing.
When you get to T(2), the zero point there's no rate of change so no current is flowing.
So that's why there's that phase difference.
Now for the inductor. As current flows into an inductor a magnetic field is created around it. The creation of that field is opposed by the space around it, and further the material that is in that space changes the amount of push back it will feel (so think iron cores etc). Now at the beginning of your sine wave (T(0) in the diagram) you are trying to change the current and the inductor is pushing back on you saying no current shall pass. The voltage is trying to push current through which is creating our magnetic field which is getting pushed back on by the space around it, so ultimately the flow of current is held up. Basically the inductor pushes back with a voltage drop that stops the flow of electrons.
Right as the voltage begins to come down in the second quarter of a cycle the field you just created begins to collapse and current flows out of the inductor (T(1) in the diagram). The ammount of current flow increases until the voltage hits the zero mark, here maximum current flows. As you continue along the rate of voltage change increases, and the inductor starts to choke off the flow of current until again you hit your maximum rate of change and there is no flow at all.
Hope that helps, also I used this other site as a reference too: http://www.allaboutcircuits.com/vol_1/chpt_15/1.html It has a nice explanation of inductors.