Is anyone aware of a computer program (Preferably free :mrgreen) that can create digital sheet music and play it back? Any help on this will be extremely appreciated. Please help! Thanks!
For tunes I use AbcNavigator2. Here’s the link.
There are quite a few out there though. I came up with loads just by Googling “online sheet music maker”. LilyPond is another that has been recommended a few times by C&F members. I have no experience of that one though.
I’ve been using concertina.net’s Convert-A-Matic for ever:
I used to have Finale, which has a lobotomised freeware version, but you can’t get the bars to line up in any comfortable way.
There are a few oddities in musescore, but I’ll back a GNU product against a Windows product unless the Windows one is FAR superior.
Musescore is okay. I don’t use it a lot.
I’d be very, very surprised if anything can compete with Sibelius (Windows & Mac) for quality, capability or usability, but obviously can’t recommend it to someone looking for a cheap/free solution!
Musescore is okay. I don’t use it a lot.
Sibelius is de facto industry standard for good reasons. I use it all the time at work and home. But it’s expensive to buy, expensive to upgrade and probably way OTT for the OP here.
“Industry”!? 
Yes, really… sure you know what I mean (or can Google it) and don’t see the need for that angry-looking smiley and stuff!
The industry began in New York in the 1850s, when they hired cheap Irish immigrant labor to crank out jigs and reels in tune-writing sweatshops under inhuman conditions, being force-fed Guinness and curry chips.
Rumor has it that it was common practice to do final tune-testing by playing them on the Irish tiompán. Hence the term, Tiompán Alley.
As Peter D. hints at, which application you want really depends on what you want to do with it. So what do you want to do with it? 
I think most notation packages can produce MIDI output, so the question is really on the notation side of things.
If you want do be able to set basically anything that a skilled music copyist could do, then you want a full-blown package like Sibelius or Finale or Lilypond. The first two are expensive; my impression is that Sibelius may tend to have a stronger following in Europe, and Finale in the US. Lilypond is free under GNU licensing; I’ve played with it a bit. It’s very capable, but it’s basically a rendering language with a fairly steep learning curve.
For basic melodies, ITM, folk music etc., ABC is excellent. There are many free or inexpensive programs that can handle ABC and produce very nice output. You can even do grand staff piano parts, multiple voices, lyrics alignment, and other fancy things. But ABC is really oriented to folk music and accompaniment, and at some point you hit the limits of the subset of notational stuff it gives you.
Of course, ABC is also the de facto standard for trad tune exchange on the Net, so it’s a good skill to have anyway. But it’s also another rendering language, typing in notation rather than drag and click. So there’s a learning curve, but nowhere near as steep as Lilypond.
Check out the main ABC page for programs and info:
For more WYSIWYG-oriented packages … I’ve heard decent things about MuseScore (free, GNU), that IB mentioned. I think Melody Assistant ($25 shareware) has a following. Finale Notepad is cheap ($10, no longer freeware, and better supported now, I think). I didn’t experience IB’s problems with it, but I prefer ABC for the same basic level of notation it gives me.
Good answers above. I must say though that there are lots of people, and lots of shops, that would take umbrage at the idea the Sibelius is the “industry standard.” Finale was there first, and still has a huge following. Some folks claim more flexibility for one or the other but I wonder…
Och well, guess they can start throwing the umbrage smileys at me too! 
I use Noteworthy Composer. Not free, but reasonably priced at $50, IIRC. Seems very capable, 'though I haven’t compared it to anything more advanced. I didn’t find it difficult to learn. Large library of contributed scores, leaning to classical, hymns, and old-time popular music.
If you go the ABC route (which is a very good one for most trad music, particularly Irish, Scottish, British and Scadinavian) check out EasyABC. Free and very useful. And, it works on most any computer. PC or Mac or even Linux I believe