How is 6/8 related?
Just like there are multiple ways of notating 4/4 songs (like 2/4, 2/2...) the same applies to songs written in 3/4 time. Depending on your tempo and how you count things, on occasion it may make more sense to notate and count the song in groups of six eighth notes, in which case you would write the song in 6/8.
Simply Red's "If You Don't Know Me By Now" is an awesome example of a 6/8 song. If you listen to the groove you can clearly count the beats as being 1-2-3-4-5-6, 1-2-3-4-5-6, with beats one and four being the accented ones. Benj Pasek's and Justin Paul's "Tightrope" from The Greatest Showman is another great song in 6/8. Seal's "Kiss From A Rose" and Billy Joel's "Piano Man" also belongs in the 6/8 category.
Using 3/4 time vs. 6/8 time
The thing to keep in mind is that the two time signatures are truly interchangeable, which is why I call them related cousins. Almost any of the above-mentioned songs could be counted either in 3/4 or 6/8. It's really a matter of how you notate things and how you want to count them. Would the lead sheet look cleaner as quarter notes or as eighth notes? It's really up to you. The underlying song will remain the same, regardless of the time signature you choose.
Here are a few more songs for you to listen to. What do you think? 3/4 or 6/8?
Eagles - “Take it to the Limit”
Rolling Stones - “Dear Doctor”
Red Hot Chili Peppers - “Breaking the Girl”
Simon & Garfunkel - “America”
Bob Dylan - “The Times They are a Changin’”
Muse - “Blackout”
Jimi Hendrix - “Manic Depression”
Metallica - “Nothing Else Matters”
John Mayer - “Gravity”
The Muppets - “Rainbow Connection”
How do you write a song in 3/4 or 6/8?
The songwriting part of it doesn't differ in any way from writing any other song. As long as the inspiration is flowing, that’s most important. But you do need to make sure that you're giving yourself a steady groove that follows your intended rhythm. If you're using Spire Studio, head over to the Tempo tab and set the metronome to either 3/4 and 6/8 time. Turning on the "click" will allow you to stay within the parameters of the time signature and will help you write a song in that meter.