What is the meaning of "equivalent weight" and its formula?
Definition of Equivalent Weight (also known as equivalent mass):

The simplest definition of equivalent mass I can think of is the following:

The mass of a compound which will combine with 8g of oxygen or 1g of hydrogen is said to be the equivalent weight of the compound.

Lets take an example,

$\ce{2Ca + O2 -&gt; 2CaO}$

From basic stoichiometry, you know that 2 moles of calcium react with 1 mole of oxygen to produce 2 moles of calcium oxide. We can say the same statement in terms of masses as 80g of calcium reacts with 32g of oxygen to produce 112g of calcium oxide. 

Since 80g of Calcium reacts with 32g of oxygen, 20g of calcium should react with 8g of oxygen.

Now by the definition of equivalent weight mentioned earlier, the equivalent weight of calcium is 20g.

$$E = \frac{M}{V}$$

The formula you were talking about gives the equivalent weight of an element (can be extended to be used with ions).  

Equivalent weight = (Atomic mass)/(Valency)

This could be intuitive and obvious for some. Something with valency of 1 would combine with just 1 hydrogen atom, or in other words, a mole of compound with valency 1 would react with 1g of hydrogen. If the element has a valency of 2, then it would love to combine with 2 hydrogen atoms, or in other words, a mole of the element will combine with 2g of hydrogen. By definition, equivalent weight is the mass of the substance that would react with 1g of hydrogen. Therefore you must divide the mass by 2 to get equivalent weight. In fact, this formula based definition of equivalent weight of an element is identical with the definition mentioned earlier in this answer.

Previously, we determined that equivalent weight of calcium was 20g using the other definition. We should obtain the same equivalent weight for calcium by using the formula.

Equivalent weight = (40g)/2 = 20g

Definition of equivalents:
Another terminology you might come across is equivalents.

Equivalents is similar to moles. The number of moles is given by (mass of the compound)/(molecular mass). Similarly the number of equivalents is given by (mass of the compound)/(equivalent weight).

For example, 1 equivalent of calcium contains 20g of calcium. 2 equivalents of calcium contain 40g of calcium.

The utility of equivalent weights is amazing. In any reaction, $N$ equivalents of any compound on the reactants side reacts with N equivalents of another reaction to produce $N$ equivalents of the product(s).

If that sounds complicated, this example should make it clear.

20g of calcium reacts with 8g of oxygen to produce 28g of calcium oxide.

An equivalent statement is, 1 equivalent of calcium reacts with 1 equivalent of oxygen to produce 1 equivalent of calcium oxide.