You probably know Google Hangouts as an app that pops up in the corner of a browser tab or on your smartphone, but it's possible to archive your messages in Gmail too—from here they can be searched, labelled and starred so you can bring up the most important ones whenever you need to.
Conversations only appear in the All Mail section of Gmail if you've ticked the Hangout history button in a conversation window: It's behind the cog icon in chat windows on Gmail, Google+ and the dedicated Hangouts site, as well as in the mobile apps. This is the option enabled by default, so it should already be activated unless you've turned it off for certain chats.
To find your conversations in Gmail, click Chats from the list of labels on the left (you may need to choose More labels to see it). You can also type “in:chats” in the Gmail search box—add a word or phrase of your choice if you want to search your Hangouts history for something specific (like an address or phone number someone sent you).
If you've got the default conversation view switched on in Gmail's main settings page, chats appear as one continuous stream; otherwise each message shows up as an independent entity. Using the drop-down menus to the side you can reply to Hangouts with an email or even forward a Hangout to another email address.
Labels and stars are probably of most use when it comes to organizing your chats and these can be applied in the same way as with emails: All the star colors you've set up in Gmail are available for chats too, enabling you to quickly highlight ones you don't want to forget. Labels are helpful for narrowing down searches too—try “in:chats label:labelname” in the Gmail search box, for example.