How would you balance the chemical equation BeCl₂+Al(NO₃)₃→BeN₂O₆+AlCl₃

Chemical reactions occur when atoms of the reacting species come together to mix and/or rearrange to form new chemical species. Since it is simply a rearrangement of atoms, the total number of atoms of each element involved in the chemical species in the reaction must remain the same before and after the chemical reaction. This also reflects the law of conservation of mass. 
To count the number of atoms of a particular element in a chemical reaction, count the number of atoms of that element in the compound involved (subscript) and multiply by the number of moles of the chemical in the reaction (coefficient). The number should be the same in both sides of the equation (reactant and product side) when the equation is balanced.
BeCl_2 + Al(NO_3)_3 -> BeN_2O_6 + AlCl_3
There are various ways to balance chemical reactions, and the technique to do it will vary from each person. Sometimes, it is easiest to look at the simpler compounds. For instance, there are two chlorine atoms in the reactant side, and three in the product. By putting a coefficient of three and two in front of BeCl2 and AlCl3, respectively, this will balance out the number of chlorine atoms to 6. Then, check the numbers of the other atoms:
3BeCl_2 + Al(NO_3)_3 -> BeN_2O_6 + 2AlCl_3
Be - 3, 1
Cl - 6, 6 (OK)
Al - 1, 2
N -  3, 2
O - 9, 6
Notice that balancing the number of Be and Al atoms will already balance the equation:
3BeCl_2 + 2Al(NO_3)_3 -> 3BeN_2O_6 + 2AlCl_3
This time, there are 3 Be atoms involved, 6 chlorine atoms, 2 aluminum atoms, 6 nitrogen atoms, and 18 oxygen atoms - the equation is balanced. 
This technique is called balancing by inspection. This technique works and requires a lot of practice.
In the same way, this can be solved algebraically, by using variables for the coefficients:
xBeCl_2 + yAl(NO_3)_3 -> zBeN_2O_6 + wAlCl_3
There are 4 variables, and a total of 5 equations (one for each atoms). N equations are necessary to solve a system of N variables so this system is actually over determined (but you don't have to worry about this).
For Be: x = z
For Cl: 2x = 3w
For Al: y = w
For N: 3y = 2z
For O: 9y = 6z
Since these variables represent the number of moles of species relative to one another, we can arbitrarily choose any of them to equal any number, and get the others based on that - and then simplify if we can - that is, reduce the molar ratios to their lowest terms, or get rid of fractions.
For instance, if we take x = 1, this will mean z = 1, w = 2/3, y = 2/3. The fractions can be eliminated by multiplying all the variables by three so that x  =3, z  =3, w = 2, and y = 2, which leads to the same balanced chemical equation derived above.
http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_2/node3.html