Current in series resistors and voltage drop in parallel resistors
The diagram shows first four resistors in series then four resistors in parallel.
For the resistors in series the current flowing into the wire, $I_{in}$ must be the same as the current flowing out, $I_{out}$ because the current can't escape from the wire. There is only one route for the current to flow through the wire so the current has to pass through all the resistors in turn. That's why the current passing through every resistor must be the same.
Now look at the resistors in parallel. The point here is that the top ends of the resistors are all connected together so they must all be at the same voltage $V_{in}$. Likewise the bottom ends are all connected together so they must be at the same voltage $V_{out}$. That means all the resistors have the same voltage drop across them of $V_{in} - V_{out}$.
I'll start with current first...
1) "Current flows in a circuit" is the simple answer. In other words - It's the rate of flow of electric charges. Other than $i=\frac{dq}{dt}$, Current is also given by $I=nAEv_d$ which says something that it depends upon the drift velocity of electrons. The drift velocity is the average velocity between two successive collisions. This velocity prevents the electrons from accelerating continuously. Ok. Let's consider a circuit with three resistors with resistances in an increasing order $R_1>R_2>R_3$.
First, current enters $R_1$. After some collisions (causes heat generation), it exits the resistor. Now, the same current enters and exits $R_2$ & $R_3$ in the same manner. One point is to notice that, the rate of flow of charges is always the same (the current entered and exited the resistors with same magnitude). Only the drift velocities vary in different resistors.
If the same are connected in parallel (Now, we look into the resistors), current flows through $R_3$ easily. Because, $R_3$ requires a lesser $v_d$ (i.e) Electrons entering $R_3$ would exit within a small period of time relative to the other two (thereby increasing the rate)... As a result, larger current would be observed.
2) Voltage is simply the "Energy per unit charge" in both electrostatics & current electricity. Let's assume that the electrons have some energy and they pass through the series of resistors.
When the electrons encounter collisions within the resistor $R_1$, they lose their potential (which is referred to as "voltage drop") accordingly with their drift velocity (i.e) number of collisions which evolves as heat (depends upon the resistance of each). The remaining potential is still present in those electrons which drops across other resistors. Thus, the sum of each potential drop gives the total potential in that circuit.
As in case of parallel circuit, current has divided. Now, though the charges are taking different paths, they have the same potential (energy). Hence, the potential drop across each resistor would be the same... Man, It's all the consequence of Ohm's law $V=IR$.
Note: And, sorry for that confusion. Electrons gain energy only as they pass through the battery. But, they're always continuously accelerated by the electric field. Also, resistance is also given by $R=\frac{mL}{nAe^2\tau}$ which Simon Ohm replaced it as a constant (for a given temperature)...