The process, for some people, is linear. Joseph gave an example of a linear approach: "idea ... title ... music ... melody ... lyrics"
If we accept that Joseph has captured the main ingredients with his nice summation, then we can dwell on various combinations of these for a while.
For example, it is possible to have "idea . . . lyrics . . . melody . . . music . . . title", an equally valid way of producing a song. In fact, as there are five ingredients, there are 5! (factorial five) combinations, i.e. 5 x 4 x 3 x 2 x 1 = 120 possible, and equally valid, ways of assembling a song in linear fashion. Some might argue that the number of combinations should be fewer, because logically, you would start with an idea first. This is not always the case. Sometimes I've had no idea for a song, but a phrase or a melody pops into my head and I build something from that, not really knowing what it is going to be about until later. However, I concede that it would be exceedingly rare (though not entirely inconceivable) to get through a sequence such as "title . . . melody . . . music . . . lyrics . . . idea".
However, as Burt (Whatmysay) notes, not all songs are created linearly. Sometimes these ingredients emerge in parallel, with the development of the lyrics running alongside the development of melody, with each influencing the other. At other times you might create a chorus, or a verse, from which you construct the basis of a song, then find the rest of the words later. Sometimes you can plunder your own resources, and marry a middle eight of an old song with one you are creating.
When Dreamsound2 states "There are two ways of composing, forwards and backwards", he is broadly right in that, quite often, a songwriter will work on a set of lyrics, perhaps alongside the melody, before determing the structure (forwards), or create the music and write the lyrics after (backwards). However, this generalisation is not very helpful, because there are so many other ways in which a song can be crafted, and in the context of the article, even less helpful, because of its didactic nature: there are two ways (no more, no less) of composing, as opposed to something that acknowledges the diversity of this art: "two common ways of writing songs are . . .".
There are other problems with the article, which I won't dwell on too much, but the main points are:
1 the art of songwriting is not easily encapsulated in three and a half paragraphs. I would balk at the task of trying to summarise it in a short article like that.
2 the article mixes up songwriting and recording techniques (para 4). Arranging a song is an artform in itself, and so is recording that arrangement.
3 and I am puzzled by this recommendation: "I also recommend a regular tape recorder which you can output back to your DAW and get the original saturation and warmth from regular tape." The purpose of getting a regular tape recorder was, I thought, to capture ideas. A casette walkman or a dictaphone was handy for this sort of thing . . . you know . . . you're driving along somewhere and you get this melody in your head . . . and it's vanished by the time you get a chance to do anything with it. There are more sophisticate devices around now for that. But for getting "the orignal saturation and warmth"? I don't buy it.