One meaning is, of course, music from the 'Classical era' - whether you see this as European 'art music' written between 1750 and 1810, or lay the chronological boundaries slightly wider, the meaning seems fairly clear.
At the other end of the spectrum, 'classical' seems refer to an approximate, and somewhat arbitrary, sociological boundary around a range of musics that tend to be associated with the wealthier classes, the educated, and the church. Beyond this, it's hard to pick out one particular indicator. One might say that classical music is written primarily for the sake of the art of the music itself, but operatic music would be seen as 'classical' and yet serves an entertainment purpose, while liturgical music also has a primary purpose other than the art itself. On the other side of the boundary, many examples of jazz music could be said to elevate the music itself to being of primary importance, although all of jazz is normally seen as being distinct from classical. (Having said that, jazz is sometimes referred to as America's classical music, so perhaps it's the exception that proves the rule).
One of the indicators of classical music is the separation between the concept of the composer and the performer, and the score as an 'interface' between the two functions. Of course scores and a separation between composer/songwriter and performer are common in genres that would be seen as non-classical, But the reverse is rarely true - it's not common to come across a classical work without some kind of score, whereas it is common in most non-classical genres to work without a score.
I would agree with WillRoss that between the narrowest and broadest meanings of 'Classical'/'classical', you could put your finger on any number of points, so there's no particular number of specific meanings you can pin down.
Can I class some 20th-century composers as "classical"?
Of course! Shostakovich wrote "Symphonies" on score for performance by orchestras in concert halls, and now has his music studied in academic circles - all of these are indicators (though not definitive ones) of being 'classical'.
A more tricky question might be to consider some of the prominent film composers of the 20th and 21st century - are they 'classical', or are they simply writing music in a 'classical style'?