Travelling to Tunisia from Manchester airport connecting in Frankfurt

The key point for your question is all on one ticket. This is generally the easiest type of connection to have. What will happen is:

1. In Manchester you check in. Unless there are special circumstances^[1] your bags will be checked in for the entire trip and you will receive the entire trip’s boarding passes. (Same applies if you use online check-in and luggage drop-off.) You go through Manchester airport,^[2] board your plane and fly to Frankfurt.

2. In Frankfurt when you arrive, you need to follow the signs for flight connections (Anschlussflüge, iirc).

   1. You luggage has been checked for the entire flight. Frankfurt airport ground staff will make sure that it moves on to Carthage without you needing to do anything.

   2. You will need to make your way to your next plane’s gate. If I am informed correctly,^[3] you don’t need to go through any passport controls. Your incoming flight is from outside the Schengen area and your outgoing flight will also go outside the Schengen area. Both of your flights — being Lufthansa and extra-Schengen — should arrive and leave from the B gates, making transferring slightly easier.^[4]

3. Finally, your second flight arrives in Carthage, you get off, clear immigration, collect your luggage and clear customs.

By the way: you can download an airport guide and a transfer guide from Frankfurt airport’s webpages.

Notes:

[1]: ‘Special circumstances’ typically applies to long, overnight layovers for which you would want to access your checked luggage and re-check it the next day. Not applicable to your layover of one hour.

[2]: You do not need to pass exit immigration. But since Manchester is not part of this question, this is just a footnote.

[3]: Caveat: I never actually transferred from non-Schengen to non-Schengen in Frankfurt; I always had a Schengen leg. But everything I read on the airports website seems to say that my assumption is correct.

[4]: Even though there are no systematic controls, an official may (randomly) be present at the gate to perform a (somewhat less thorough) check of documents, as noted by phoog.

I tried checking the website for Frankfurt's airport but all it says is:

In most cases, your bags will be forwarded to your final destination. Please inquire when checking in to make sure this is the case, while mentioning that you are only changing planes in Frankfurt. A sufficient number of luggage carts are available in both terminals.

Generally speaking though if you fly with the same company, your connecting flights are on the same booking, and are transiting in an airport with an "international area" (i.e. airports outside the USA, for example) then the luggage should be forwarded on the connecting flight without you having to touch it.

Just to be on the safest side possible, though, you might want to call Lufthansa and ask. They will definitely know since Frankfurt is their main hub. You can also verify upon chech-in by looking at your checked-in luggage tags: they should be stamped with the final destination airport.

On a different note, and I am not saying this to scare you, consider that a 50 minutes layover might be short for handling luggage between international connecting flights. Therefore, keep in mind that your suitcase might arrive to Carthage on the following flight, and pack a change of clothes in your carry-on. ;)