Hi and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
The problem is, Apple has replaced iPhoto with new application called Photos app from OS X 10.10.3 onwards. You can open Photos app from Applications folder. 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.
See here --> http://www.theverge.com/2015/4/8/8367469/apple-photos-app-launch-iphoto-replacement
The only iPhoto that support Mac OS X 10.10.5 is iPhoto 9.6.1. It seems like you didn't upgrade to latest iPhoto prior to the OS X upgrade you did. That it is why it is not working now.
So first, I want to check whether you are eligible for iPhoto 9.6.1. Open your Applications folder and do you see iPhoto there ? If so, 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 or later
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. Sorry.
Remember, Apple will eventually drop support for that version of iPhoto as well. So I'd recommend to start using the Photos app. So you will not face any compatibility issues in the future.
When you open Photos for first time, it will ask you to migrate your iPhoto Library, so do it. If it doesn't migrate, then do the steps below to transfer them to Photos. Hold down Option key and open Photos. Select Choose Library and locate and select your iPhoto Library.
If that didn't work, then you need to manually import the photos from iPhoto Library to Photos Library. Follow the steps below
Hold down Option key and open Photos.; Create New Library.
Then go to your Pictures folder and you will see your iPhoto Library in there. Right click (or Control + click) on the iPhoto Library and select Show Package Contents. You will see Masters or Originals folders in there. Just click and drag them to the Photos icon on the dock and they will be imported to Photos.
Please let me now if you have any difficulty, so I will continue to help you.