There are various quite complex (and mathematically precise) ways to think about degrees of freedom. However, for most purposes when analysing data it means the number of independent data items minus the number of values you have fitted to the data. For example, if you have 100 (x,y) data items and then perform a simple linear regression the software will fit a slope and intercept of the regression line – that is you have fitted two numbers, hence the remaining degrees of freedom is 100–2 = 98.
Informally (but very close to the vector space mathematics) you can think of this as having 100 pieces of randomness to start with; two of these are used up in the two fitted values, so you have 98 left.
