Tags and mentions are a way to engage with people on Instagram. They're not the same as hashtags.
A tag is usually used when someone (or their product) is in the picture/video that you're posting. But you can also use it if your post is just talking about someone/their product. You tag people when you post the picture/video. See pic below.
Mentions can be used for the same reasons but they are added differently. You add them in the captions or to your story using the "@" before the account that you want to mention.
How Is All Of This Different From Hashtags?
Hashtags serve a completely different purpose on IG. Hashtags are for searchability. They help your posts to be found be people that don't follow you. Tags and mentions are about establishing and building relationships with people on IG. These relationships help your account to grow, especially if they're two-way relationships.