Kirchoff's current law and Kirchoff's voltage law

Kirchoff's current law also known as kirchoff's point law or Kirchoff's junction law. It states that total algebric sum of current flowing to / away from a junction (point) is equals to zero. That is at any point in a circuit total incoming current is equal to total out going current. With the help of this we can find out the current in a branch of circuit if other branch currents are known.for example  if  a point has three branches and incoming current from branch 1 is 4 ampere. and out going current from branch 2 is 1 ampere. then the current from banch 3 is 4-1 = 3 ampere out going.

Kirhoff's voltage law or mesh law. it states that in any closed mesh (circuit) the algebric sum of all voltage rise or drop is equals to zero.

Voltage rise:- in a circuit if we move through cell in negative to positive direction it is voltage rise and  from positive to negative it is voltage drop.

In other circuit elements if we move in the direction of current flow it is considered as voltage drop. if we move in direction opposite to current flow it is considered as voltage rise.

Cloes circuit or mesh:- In any complex circuit we can divide it in to many closed loop. A closed loop is any path in circuit in which we start from a point and return back to same point.
with the help of using these two laws the current flowing in all branches of network can be calculated easily. A circuit diagram is given below.

In the circuit diagram shown above we can apply kirchoffs current law at any point. at poit B there are three current ie I, I1 and I2 according to KCL 

I = I1 + I2  --------- (1)

Similarly at point E

I = I1 + I2  --------- (2) n 

There are three loops (mesh or closed circuit) in diagram

first loop ABCDEFA in cyan color for which voltage equation is

V = I1R1 + I1R2 + I1R3    --------- (3)

Second loop BGHKEDCB

in this loop there is no voltage source. we start from point B and complete the loop in clock wise direction. When we move in direction of current there is voltage drop and we give it a negative sign.

When we move in a direction opposite to direction of current there is voltage rise and we give it a positive sign.

In resistance R4 and R5 there is voltage drop. in resistamce R1 R2 and R3 there is voltage rise so equation is

V4 + V5 - V3 -V2 - V1 = 0  ------(4)

I2R4 + I2 R5 - I1R3 - I1R2 - I1R1 -----(5)

Third loop ABGHKEFA

V - V4 -V5 = 0  -----(6)

V = V4 + V5   -------(7)

V = I2R4 + I2 R5  ------(8)

With the above equations value of I1 and I2 can be determined.

Above is simple circuit but any circuit can be solved.