Change a field in a different table by using a multiple choice option

You can do this with Actions.  Setup an Action on the table where the multiple choice field is.

You will need to have a Report Link field built which defines the set of record(s) to be updated.

Yeah I tried to do this. I'm not familiar with Quickbase actions. I tried to edit the related record based on the report, but the fields still don't change.

You could also pass  that checkbox field from table A as a lookup field to table B, and then have a formula field in table B to inspect that and incorporate it into your formula for setting the field in table B.

I don't want to be forced to select the record I want if I pass it as a lookup field to table B. I thought about doing that, but the added step isn't worth it. 

Not sure I understand your response. You say you have a relationship between these to tables. So adding an additional lookup field will take very little effort. If you want its value to be available in table B, why do you even need a separate field in table B which is essentially the same thing. If you were to use an Action, you would effectively be forcing yourself to use that value from table A just the same. If you have additional logic that needs to get incorporated into the setting of table B fields, you can change the destination field to be a formula that can incorporate what's getting passed from table A's lookup field, along with any other fields in table B.

I have a relationship, therefore, I have to select which record I want to show in Table A. I don't want to go and edit the related record and choose which record in Table B I just created. So a little background, I am creating a new record in table B (by clicking a button in table A) and I want the information from this created record to automatically update back to the related record in Table A. Table A will not see this until I select the brand new record from the multiple choice list.

I'm confused now. This scenario sounds different from what you originally asked in this post.

Let me restate this. I have a table that holds a bunch of contracts. Within each contract, there is an button that will allow you to add a document in the documents table (separate from the contracts table). When you click the add document button, it will take you to the general main form for the documents record. When a specific document is uploaded using the file attachment field, a checkbox will be filled in saying that type of document was uploaded. When the document record is saved, it will take me back to the contracts record. Back in the contracts record, I want this checkbox to show up saying that this type of document was added when clicking the button. I want the contract record to somehow show that a specific type of document was uploaded. How do I do this or is there a better way of doing it? 

Quinten, 
You are way over complicating this now that we under stand your question.  Had you stated your actual use case we could have more easily helped you.

Just make a summary field on the relationship to count the # of document of a particular type.  say Type = Liability.

Then make a formula field on the parent Contracts record called [Contract Documents Loaded for Liability]

IF([# of contract documents for Liability] > 0, true) 

I thought about that. The problem is that more than one document is going to be added for that related contract. However, only one document is going to be the "liability" one. I have a multiple choice field with the many different types of documents. How do I specify that I only want the "liability" ones counted?