Would it be trademark infringement because I use a common word - "Memory" - that a company says they have trademarked?

First of all, don't post specific details like actual names regarding your potential trademark dispute on this public site - this huge company with a legalo department may be monitoring these sites.

Secondly, yes, many trademarks exist with the word "memory" in it.

Whether they've actually trademarked the word for their product and/servcesis something that has to be verified. And whether your use infringes on their use is a fact-specific inquiry that depends on who's used what, for what, since when, and where and how, which is a long way of saying "it depends."

Presumably they trademarked the word with other words as well, some words of which may be disclaimed by the registrant, meaning they're not claiming rights to all the words. The fact that there are other alleged infringers out there cuts both ways - it doesn't make you less culpable for your infringement, if it is infringement, but the existence of so many other users does weaken the mark which makes a finding of your infringement less likely. And the word itself is descriptive of computer games anyway, which also makes it a weak mark.

Ideally you'd go to an IP lawyer and fully and accurately dislcose your situation and get some specific advice about how to respond to this "cease and desist" letter.

But as a practical matter, big companies often try to muscle out smaller or newer companies with weak claims in the hopes that they won't get any meaningful push back. If you can't afford any legal help, it's not going to matter much whether this big German company is in the right or not if they're willing to sue you and you're not willing or able to fight back. You can argue that their mark is weak and should be cancelled, but you should get a lawyer to argue this. This may result in you being sued, so you need to decide if you want to fight this or not.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

I agree with my colleague. Without actually seeing your mark and the other company's mark and their trademark registrations there is no way to give you good advice on how you should proceed. You should speak with a trademark attorney for further guidance.

I've had this exact situation come up for clients in the past. The answer depends on exactly what registrations the company that sent the letter has, and what your mark is, and your product. I agree with my colleagues, this is not a situation where a public forum will provide you with a useful answer. You will need to speak to a trademark attorney.