I have been playing the flute for some time now, and wanted to expand my knowledge of it. To do so, I started learning more about the standard music notation. While doing so, I ran into a concept called chords. I learned all about chords, and desired to play them on my flute, but couldn't think of a way to accomplish this. Is there a way to play chords on the flute?
The flute, like most wind instruments, is considered to be monophonic (as opposed to a piano, which is polyphonic), meaning you can only play one note at a time (within reasonable ability).
However, there exist "extended techniques" on the flute that go beyond the standard teachings. In this case, "multiphonics" (which is an odd term, since "multi-" is Latin while "phone" is Greek) allow you to play dyads (i.e. 2-note combinations, otherwise known as intervals) and even chords (3+ notes), based on the properties of harmonics and overtones.
The trouble is, these are very difficult and only possible with a certain selection of notes (depending on the fingerings), and you're probably not going to get a very pleasing tone on them.
There are several resources on this topic you can try to use. I'm not skilled with multiphonics, but For the Contemporary Flutist's suggested technique is to imagine your tone being a vowel - higher tones shape your embouchure like an "e", while lower tones need you to form an "o" - and try to play with two vowels at once, one at the top of your mouth and one at the bottom. The Virtual Flute suggests possible fingering combinations for optimal multiphonics.
It should be added to the other answers that in order to create in your ear the impression of playing a chord, the notes which make up a chord don't necessarily have to be played at the same time. They can as well be played as a sequence, which is called a broken chord or arpeggio. If it sounds pleasant to accompany a song playing the broken chords on a flute, I don't know, but you can easily use this technique if you just want to hear what a chord sounds like.
Just to add to @Bladewood's answer, practising harmonics will help. Start with "bugle calls" by fingering a low D, say, and trying to hit higher notes. These higher notes are the harmonics. You should be able to get 4 or 5 notes out of that single fingering with a little practice. THEN aim to hit the first two together, then the first three.
There are good reasons to practise harmonics other than trying to make chords. It helps to project your lower octave, and if you’re clever with the harmonics, it can give you alternative fingerings for tricky passages, too (only for emergencies, though! There will be tuning issues to overcome). It also help lip flexibility.
If you have a reasonable singing voice then of course you can sing whilst playing a different part. With practice you can produce two part harmony and even two part counterpoint.
If you choose the singing pitch to coincide with the harmonic you want then this will resonate within the flute and make the harmonic easier to produce. Of course there is a limit to this last idea depending on the range of your voice.