While there are some special exceptions, most radio communication is usually achieved with the aid of upconversion and downconversion.
Basically, a transmitter starts with a circuit for modulating information (be it voice or data) onto a signal of conveniently low frequency that is easy to work with - a few tens or hundreds of kilohertz for narrowband applications, often somewhere between 10 and 45 MHz for wider band ones. At these frequencies analog circuits work well, or one can actually use a D/A converter at the output of a DSP that does the modulation mathematically. (For data rates higher than a "DSP chip" can handle, parallel logic in an ASIC or FPGA is used, so each individual path may only have to calculate every 8th or 32nd or whatevereth sample needed by the DA).
The transmitter also contains an oscillator or synthesizer for generating a signal closer to the desired transmitter frequency, and a mixer which multiplies the two signals together, causing generation of sum and difference frequencies. Either the sum or the difference will be the desired transmission frequency, and is selected by a filter, amplified and sent to the antenna. (Occasionally multiple stages of conversion are needed)
The receiver works the same way, only in reverse. A local oscillator signal is subtracted from the amplified antenna signal (or the other way around), creating an intermediate difference frequency that is back down in the range more convenient to work with (In AM broadcast receivers, typically 455 KHz - for FM, traditionally 10.7 KHz and then converted down againto to 455 KHz, though today staying at 10.7 MHz works too). This intermediate frequency can be processed by a demodulator circuity, or digitized in a fast A/D converter and fed into a potentially parallel DSP to complete the process.
If the desired bandwidth of data to be transmitted is less than about 10 KHz, one can actually use a computer soundcard to make a high performance receiver or transmitter, by positioning the intermediate frequency at say 10 KHz and using software to process a bandwidth spanning 5-15 KHz.
Today a common technique is to exploit some properties of complex numbers, and do the modulation/demodulation balanced around a center frequency of 0, such that it contains both positive and negative frequencies. By using two phases of the oscillator and something called an image reject mixer, one of the two resulting frequencies cancels and the other reinforces. However, two D/A or A/D converters are needed - one for the "I" phase and the other for the "Q". You can sort of do this with a stereo soundcard, though the DC blocking caps will create a hole in the passband right in the middle, at what gets converted to 0 frequency.