When you balance an equation, you want an equal number of atoms on both sides. Let's begin with Fe, or iron.
Fe2O3(s)+CO(g)→Fe(s)+CO2(g)
There are 2 atoms of Fe on the left but only 1 on the right. Add a coefficient of 2 to the Fe atom (or multiply Fe by 2) on the right to get 2 Fe atoms on the right side of the equation:
Fe2O3(s)+CO(g)→2Fe(s)+CO2(g)
Now let's focus on O atoms since they are next in the equation after Fe. There are 4 atoms of O on the left side of the equation, while only 2 on the right side, so add the coefficient 2 to the O on the right side (Note: this changes the number of C atoms as well):
Fe2O3(s)+CO(g)→2Fe(s)+2CO2(g)
We need to fix the number of C atoms on the left now, so add 2 to the CO to make the number of Carbons equal (This changes the number of O atoms too):
Fe2O3(s)+2CO(g)→2Fe(s)+2CO2(g)
There are now 5 O atoms on the left with 4 on the right, and we can't add a fraction as a coefficient. This means we have to change the numbers so that the numbers of C atoms are equal on both sides of the equation. Let's change the coefficients of the compounds with C to 3:
Fe2O3(s)+3CO(g)→2Fe(s)+3CO2(g)
There we go! All of the atoms on both sides are now balanced.