To "give someone information" is another possibility here.

"I will apprise you of the time" means I will tell you the time.

Once the secretary knows which conference room the meeting will be in, she will apprise, or tell, everyone.

Alex asks everyone to sign their names on a sheet of paper before leaving.

"To sign one's name" means to write your name on a piece of paper to show that you have read or agreed to something.

We typically sign checks from the bank, for example, or letters, or maybe contracts.

In this case, the people at the meeting are going to sign a piece of paper.

We call that piece of paper a "sheet" (sheet).

A "sheet" of paper is one piece of paper; in most American offices it is 8! x 11 inches - that's the size of the paper, but a sheet of paper can be any size.

Alex says that the signed sheet of paper will give them a record of who was at the meeting.

In this case, a "record" is something that is written down so that you can remember it later.

The sheet of paper that Alex is talking about will provide a record, or a way of remembering, who was at the meeting.