I believe I understand how tuning a radio with an analog tuner works: turning the dial physically changes the length of the antenna, which determines which broadcast wavelength will resonate in the antenna and get picked up. In contrast, what is the mechanism that makes a digital tuner work? My guess is it'll be some clever little circuit that somehow selects a resonant frequency by changing the amount of current going through it (or something like that).
In the early days of radio, the resonance of the antenna in combination with its associated inductive and capacitive properties was indeed the item which "dialed in" the frequency you wanted to listen to. You didn't actually change the length of the antenna, but by changing the inductor (a coil) or capacitor connected to the antenna you tuned the resonance. The output signal is an alternating voltage, and by rectifying it with a diode (called a "crystal" then..) you could extract a signal modulated as a varying amplitude of the carrier wave. All this without any battery! :)
But actually the antenna in a normal modern radio is not the component that "dials in" the selected broadcast frequency. The antenna circuit should indeed have a resonance within the band of frequencies you are interested in but this wide-band signal is then mixed with an internally generated sinusodial signal in the radio in an analog component, this subtracts the frequencies and lets the rest of the radio operate on a much easily handled frequency band (called the intermediate frequency). It is in the mixer you tune the reception in a modern superheterodyne radio receiver. It is much easier to synthesize an exact mixing frequency to tune with than to change the resonance of the antenna circuit.
The rest is not really physics, but the difference between an analog and a digital radio comes in the circuits after this and basically an analog radio extracts a modulation from the intermediate frequency which is amplified and sent to the speakers or radio output. In a digital radio, the signal represents a digital version of the audio, just like a WAV or MP3-file on a computer is a digital representation which can be turned back into an analog signal you can send to a speaker. The benefit of this is that the digital signal requires (potentially) less bandwidth in the air so you can fit more signals in the same "airspace" and that the digital signal can be less susceptible to noise. I write "can", because unfortunately many commercial digital radio/TV stations don't do this to improve the viewing or listening quality but just to fit in more content.
Let me reiterate that in a "digital" radio, the component that selects the reception frequency is still analog but the mixing (tuning) frequency is digitally controlled and selected.
There is also a very interesting thing called Software Defined Radio, SDR, which is the principle where the intermediate frequency (or in some cases the antenna frequency directly) is turned into a digital signal and demodulated by a signal processor which is completely software-upgradeable. Since it is much easier to program new software than to solder electronic components around, this created large interest in the radio hobby community where you can completely change the properties of a radio receiver just by downloading someone else's software from the net or write a new one yourself.
If you include SDR, and apply it without any intermediate frequency (take the antenna directly to an analog/digital converter and into a signal processor), you do indeed have a purely software-way of tuning your source like you ask for, although this is not how the most common digital radios work currently.