Reading "Understanding DSP" of G.Lyon .Actually,DFT is sampled function of FT in both time and frequency domain.Because of that reason,DFT can be calculated by processor or computer
DTFT is used to transform a discrete function from time domain to frequency domain.
But the spectrum you get by applying Discrete Fourier Transform DTFT is continuous.The DFT will produce a sampled (discrete) version of the DTFT specturum (i.e something you can implement using a digital system with small amount of memory).