If the signer has signed the document, but the status of the transaction in the overview page has not changed from e.g. Started to Completed, there is a great chance that the signer forgot to click SEND after specifying the signature.
When the signer has signed the document a window will appear, it is seen below.
It is important that the signer clicks SEND. Otherwise, this transaction will not be completed. If the transaction entails more signers, the next signer won't receive the invitation message.
Check the Activity Log
If you are not certain that this is what happens in your case, you can check it in the activity log. If the signer claims to have signed all documents, there is a good chance that he/she forgot to click SEND.
When the recipient has clicked SEND, following line will be added in the transaction's activity log saying "All documents have been signed by Recipient Name":
If this line has not been added to the activity log, the recipient needs to open the e-mail once again, and here he/she will be meet by the pop up window allowing to click SEND and thus, complete the signing.