ECP-Equivalence Class Partitioning 

Concepts: We define "Equivalence Class Partitioning" as a
method that can help you derive test cases. You identify
classes of input or output conditions. The rule is that each
member in the class causes the same kind of behavior of the
system. In other words, the "Equivalence Class Partitioning"
method creates sets of inputs or outputs that are handled in
the same way by the application

A more generalized definition "A technique in black box
testing. It is designed to minimize the number of test cases
by dividing tests in such a way that the system is expected
to act the same way for all tests of each equivalence
partition. Test inputs are selected from each class. Every
possible input belongs to one and only one equivalence
partition"

Why ECP? This method drastically reduces the number of test
cases that are required to be tested because we don't have
time, money or manpower to test everything. In addition, it
can help you find many errors with the smallest number of
test cases.

How to use ECP? 
There are 2 major steps we need to do in order to use
equivalence class partitioning:
1. Identify the equivalence classes of input or output. Take
each input's or output's condition that is described in the
specification and derive at least 2 classes for it: 
a)One class that satisfies the condition – the valid class. 
b)Second class that doesn't satisfy the condition – the
invalid class. 
2. Design test cases based on the equivalence classes. 
 
Example 1

In a computer store, the computer item can have a quantity
between -500 to +500. What are the equivalence classes?
Answer: Valid class: -500 <= QTY <= +500
                Invalid class: QTY > +500
                Invalid class: QTY < -500

Example 2
In a computer store, the computer item type can be P2, P3,
P4, and P5 (each type influences the price). What are the
equivalence classes?
Answer:         Valid class: type is P2
                Valid class: type is P3
                Valid class: type is P4
                Valid class: type is P5
                Invalid class: type isn’t P2, P3, P4 or P5 
BVA-Boundary Value Analysis
Concept:-BVA is a method that tests the boundary whether
it's an input, an output or a performance boundary. Our
tests focus on the boundaries values and not on the entire
range of data. We will use it when we have a field that can
contain a range of values as an input, an output or as a
requirement.
How to use?
If you have a range: (a to b), you will test the following:
1. a-1-invalid
2. a-valid
3. a+1-valid
4. b-1-valid
5. b-valid
6. b+1-invalid

but according to ISTQB we only test following:
1. a-1-invalid
2. a-valid
3. b-valid
4. b+1-invalid

Why ISTQB has less test cases, because we can say that if
'a' and 'a-1' are working well, then we can assume that
'a+1' is working well. The same claim will be about the
upper bound. If 'b' and 'b+1' are working well, then we can
assume that 'b-1' is working well.

Here are some more rules from my experience that you can
take in consideration: 

Always test 0 if it is inside the range and sometimes even
if it's out of range because 0 has special effect on
software (like dividing in 0). 
Always test the empty string if it is inside the range and
sometimes even if it's out of range. 
Sometimes you can test a value that exists inside the range
and not in the boundary just in case…(It allows you to sleep
deeper at night…). 

****now im really tired....LOL****

Superb explanation vivek gupta, keep it up.