Nomenclature varies, and is often used sloppily by people who assume you know what they are talking about, but you can do fourier analysis with 4 types of transforms:

1) Fourier Transform. Continuous and unbounded. Integrate[f(x) * e*(ikx), {x, -inf, inf}] The input is defined for all values of x, and the output is defined for all values of k. This would be considered the "normal" fourier transform by physicists and mathematicians because it is a map from regular functions to regular functions, thus it is the easiest to do analytically, and represents real world analog signals.

2) Fourier Series. Continuous but bounded. Integrate(f(x), e*(i k_i x), {x, 0, N}). The input is a continuous function over some interval and is effectively treated as periodic in that interval. The output is a series of sin/cos amplitudes index by integers (i.e., discrete and unbounded)

3) Discrete time Fourier Transform. Discrete but unbounded: Sum[f(x) e*ikx, {x, -inf, inf}] The input function f(x) is defined at discrete sample times. The output is a continuous function of 'k', but is completely specified by the first nyquist band, that is, k < 1/sample rate, or equivalently on the interval (- 0.5/sample rate < k < 0.5/sample rate).

4) Discrete Fourier transform. Discrete and bounded. Sum[f(x) * e^(i * k_i * x), {x, 0, N}]. Both the input and outputs are discrete and bounded (or periodic). All numerical computation is done with the discrete Fourier transform since computers have finite memory, and real data is sampled at discrete intervals anyway. Also called the FFT (Fast Fourier Transform) after the family of algorithms that makes it practical to compute.

Type 1 and 4 are their own inverses, while 2 and 3 are inverses of each other. Which one you use depends on your application. Very often what you want is to do a DFT in such a way that it approximates the continuous Fourier Transform. You can do this by making sure that your signal is limited in both frequency and time. You limit by frequency by analog filtering before digitization. You limit in time by applying a window function -- multiplication by a function that goes to zero at the beginning and end of the sample interval. If you do that, the discrete fourier transform will be a close approximation of the continuous Fourier transform. This is the big reason people tend to use the terms somewhat interchangeably: under the conditions you normally want to operate, they provide the same result even though they are technically different.