Hi and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
The problem is, you have updated your OS X software to latest version of Yosemite and Apple has replaced iPhoto with new application called Photos app in Yosemite 10.10.5.
You can open Photos app from Applications folder or from dock. Photos app does support your old iPhoto library. So you will be able to open Photos and see all the photos that were in iPhoto. To do that, hold down Option key on your keyboard and open Photos. Select Choose Library and locate and select your iPhoto Library (iPhoto Library is normally saved on the Pictures folder in your home folder).
The only iPhoto that support Mac OS X 10.10.5 is iPhoto 9.6.1. It seems like you didn't update to latest iPhoto prior to the OS X update you did. That it is why it is not working or showing now.
So first, I want you to check whether you are eligible for iPhoto 9.6.1. Open your Applications folder and do you see iPhoto there ? If so, open it. If you are unable to open it, trash it.
(You can open Applications folder from the left panel in Finder or from Go menu in the top bar)
Don't empty the trash. Then open App Store application from Applications folder. Click Purchases tab and do you see iPhoto application there ? If so, Install it. This will install the iPhoto version 9.6.1 which is compatible with Yosemite 10.10.5
Then open it from Applications folder. If you don't see iPhoto under Purchases, then you will not be able to download iPhoto 9.6.1. You have to use Photos app from now on. This is not your fault. It is Apple show intentionally removed iPhoto support and replaced it with Photos.
I'd recommend to start using the Photos app from now on instead of trying to get iPhoto from App Store. Even if you're able to get iPhoto from App Store, it will be completely removed in the future updates.
If you have any difficulty or if you have any questions, please don't hesitate to ask.