Claim or proof of claim: A proof of claim must be filed in order for a creditor to be paid. Usually no claims are filed in a chapter 7. In a chapter 13, the court gives a deadline for filing claims. After that the creditor may file a claim, but the debtor may ask the court to disallow the claim because it is "out of time" or too late to be considered. Claims are deemed allowed (i.e., they can be paid) unless the debtor objects.
Claims register: A docket kept by the court which shows officially, who filed for what sum of money. Anyone can inspect this docket. It is very useful to determine whether objections should be filed.
Motion: Simply put, a motion is a request for an order by the court. An attorney can ask the court for anything (ok, you cannot get Seinfeld back on the air). The court can bring a dismissed case back if you have a good reason (e.g. missed payments because of loss of employment, illness, etc.) The court can do other things as well. For instance, if a creditor obtained relief from the stay (below) the court can reinstate the stay.
Relief from the stay: A secured creditor (your mortgage company, for example) can ask the court to allow it to go ahead with a mortgage foreclosure if you do not make current payments. This is called "relief" because the court is lifting the stay to allow this creditor to move against your legal property interests. Creditors need a very good reason to obtain "relief." Unsecured creditors virtually never get this. I have personally never seen this occur.
Objection: A creditor, the trustee, the IRS, or any party in interest may object to the confirmation of your chapter 13 plan. If this occurs, the court may not confirm your plan if the objection is not withdrawn or overruled after a hearing.