This app was mentioned in 6 comments, with an average of 2.17 upvotes
Well Empire Legends is an android/chrome crossplatform app, its free for character but might still be pay for ship info.
There is a pretty good app in the Google Play app store that will randomly generate NPCs with traits and a small backstory that you can use for ideas. It can be found here: https://play.google.com/store/apps/details?id=com.aerilys.empire.legends.android
By the way, this is a great idea.
There's this app. It's pretty good, but it is missing a few things to make it great. It doesn't support lightsabers or the force, and you can add custom skills but you can't tie them to an ability so it won't calculate your dice correctly.
One thing I've found for doing this is the Empire Legends android app. The base version is free, and it can have all your characters/npcs on it, including things like skills, dice, strain/wounds, etc. The paid unlocks let you include ships and randomly generated NPC's (for when your players start to leave your storyline XD ) Link: https://play.google.com/store/apps/details?id=com.aerilys.empire.legends.android
Edit: Oh yeah, and I can't remember if the cloud saves are free or not, but they have them! :end edit
Disclaimer: I haven't actually played this game (or any tabletop rpg for that matter). I've been researching it for the past two weeks, and it seems like a really cool game/system. That being said, I will likely get this app when I get the game, because I think it'll work as well as pen/paper, and I'm generally a techie guy anyway.
It's very different, in that the players have a LOT of input regarding the world and events.
The advantage/disadvantage mechanic allows shades of grey to everything, so it can really flex your improv muscles (and your players as well).
Miss your attack roll but get an advantage? You blaster fire whizzes past your target's ear, missing his face. However, the electrical box behind him isn't so lucky. A shower of sparks blinds your target (this would give the next person to attack this target a boost die, or give the target a penalty on his next attack roll).
This example extends to everything, including social rolls. There are some easy ways to handle advantages/disadvantages so you aren't always thinking on your feet, but a great thing to do is ask the players what they think happens and then narrate it. This really gets the players engaged and offers a level of involvement not found in many other RPGs.
Everything else you should be able to draw parallels to D&D (difficulty die are like DC, skills are tied to characteristics like agility, talent trees, etc). The biggest difference I have found is with the Advantage/Disadvantage die. Get comfortable with those, and encourage your players to get comfortable with them, and your game will progress much smoother.
Here are some resources. Many of these will be covered by the intro box rules, and I strongly suggest you follow the way they introduce mechanics slowly. However, once a mechanic (like combat) is introduced in it's entirety these can be useful. Some of these you may not touch during the intro box, but if you decide to wing anything beyond the intro box or take your game further these may be helpful:
(FFG Star Wars Index)[http://swrpg.viluppo.net/]
(donjon generator, mainly for names but other great tools here too)[http://donjon.bin.sh/scifi/name/#type=cyberpunk;cyberpunk=ModernMale]
(FFG Star Wars Cheat Sheets for GMs)[http://thealexandrian.net/wordpress/37660/roleplaying-games/star-wars-force-and-destiny-system-cheat-sheet]
(The best way to briefly explain the dice mechanics)[http://www.uptofourplayers.com/comic/edge-empire-rulez-pt-1/]
(Campaign Podcast)[http://oneshotpodcast.com/category/podcasts/campaign/]
Added the last one because it is a group of improv actors learning and playing the system. They come up with fantastic examples for the Advantage/Disadvantage mechanic as well as just being hilarious in general. It starts a bit in the middle of things, but stick with it as they learn the system (also useful to see things your players may get hung up on) and it's very rewarding.
edit: formatting
This app might be helpful:
https://play.google.com/store/apps/details?id=com.aerilys.empire.legends.android
It is mainly for PC's though