The first Elite game is written by David Braben and Ian Bell, released in 1984. The Amstrad CPC version was written from the ZX Spectrum port. It has less ships (no Ananconda nor Transport) and minor differences in missions and titles.
Frontier Elite 2 was released in 1993 on Amiga, Atari ST and ms-dos. It's an open world without end nor scripted missions. As stated on the wiki page, it's now a shareware that you can download on Frontier Astro or its new version with better graphics working on actual PC : glFrontier.
If you want to play again Elite 2, you will find informations on Sharoma.
Frontier First Encounters was released in 1995 only for ms-dos, it was out too early with bugs which where corrected later. This time there a a story which you follow through events and missions, you will also meet the famous thargoids. Some missions can be done only with certain conditions or a minimum combat rating. It is also now a shareware which you can download on Frontier Astro with also a newer version working on current PC : Frontier First Encounters D3D. There is also glFFE and another version of glFFE by Nick.
And of course there is also Elite Dangerous which was released in 2014 on windows, MacOS, Xbox and PS4.