Term "local oscillator" (=LO) is historical. Radio equipment had initially an oscillator as the transmitter. Having an oscillator also in the receiver (to shift the signal to IF in the mixer) needed a new term and that became LO.
Let's have a signal named A. Let's input A to a mixer and let's simultaneously input from an oscillator (=from the LO) a sinewave which has frequency Fo. If the mixer works ideally, one can separate with spectrum analysis two simultaneous output signals. Every moment the voltage in the output is the sum of these mixing products:
1) frequency sum: every spectral component of A is shifted amount +Fo upwards in the frequency scale
2) frequency difference: every spectral component of A is shifted amount +Fo downwards in the frequency scale. If the resulted frequency is negative, the minus sign can be omitted.
Ideal mixer is actually a voltage controller amplifier or attenuator, where the local oscillator signal controls proportionally how much A is amplified or attenuated.
Term up-conversion isn't as clear as it can at first seem to be. Amplitude modulated radio signal contains simultaneously both ideal mixing results. If the transmitted audio signal covers frequency range say 100Hz...5kHz and the "LO" signal is at 1MHz, the mixer outputs simultaneously the audio signal at bands 1000100Hz...1005000Hz and 995000Hz...999900Hz.
To be exact an additional DC component (=0Hz) is added to the audio signal and that generates an additional mixing product, the carrier at 1Mhz, which is needed to make the conversion back to audio (=the detection) easy in AM receivers. We say the AM radio signal has frequency 1MHz although it's distributed between 0,995MHz and 1,005Mhz.
The carrier and one of the copies of the signal above or below the carrier is theoretically unnecessary. They waste both signal power and frequency space. We have SSB (=single side band) radio systems where the unnecessary parts are suppressed with filters or clever phasing circuits, but they make the receiver complex and degrade audio quality, so the simle AM radio signal is still in use. For analog television a version of AM which radically reduced the unnecessary parts but transmitted them as band-limited was used.
In digital signal processing ideal mixing with multiplication is used widely. In analog circuits perfect multiplication is possible only with low frequency op amp circuits. In radios bad mixing was so serious problem that some workarounds were needed. The most common workaround was to replace multiplication with switching.
With spectral math (=trigonometric equations or Fourier transforms) one can show that the sinusoidal local oscillator signal in ideal mixing can be replaced by square wave. The result is like the signal A is either switched ON-OFF with switching frequency=the LO frequency (=Fo) or the polarity of A is swapped with the LO frequency. We skip the math, but if we believe the the frequency shift in the ideal mixer, we see that the switching creates the mixing results also between A and the harmonics of the square wave at freq Fo.
In practical mixers no linear multiplication is attempted, the mixers do the switching with diodes or transistors. The voltage of the LO signal is so high that the switching is possible. Signal A gets thru half of the time in balanced mixers or all the time in double balanced mixers, but half of the time as inverted (=multiplied by minus 1)
The unwanted mixing results with the harmonics of the LO signal are filtered off. The system designer has selected the used frequencies so that filtering is possible.
About IF: Radio receivers need often high signal amplification to be able to receive weak signals and as well they need sharp filtering to receive only the interesting radio station. These things can be too expensive with variable frequency circuits except if the interesting station makes an ovewhelming strong signal which covers the others which simple receivers would pass through at the same time.
Mass produced "Super Heterodyne" receivers as early as in 1930' shifted with a mixer the interesting signal to about 450kHz intermediate frequency, where the final filtering, amplification and detection to audio happened. Shifting to 450kHz was actually shifting upwards when the signal was at Long Wave band. MW and HF bands needed shifting downwards.
Direct shifting with mixing to 0Hz or direct conversion receiving is possible with phase locked or synchronizing oscillation analog circuits if there's at least somehow helping bandpass filtering in front of the conversion. Also digital signal processing can be used if there's some amplification, signal level normalization and filtering available before the AD conversion.
IF has also been used in radio transmitters to make modulations easier. It's especially useful in microwave range, where everything complex is difficult at the final transmission frequency.