Carry on

Specifics as far as size and weight depend on the airline. In general, it means something that fits under the seat in front of you. A backpack (daypack size, not a luggage backpack), or a purse, or messenger bag/tote bag. I like to put my medium size purse inside a slightly larger tote bag for the flight - so I have my purse stuff plus a little extra room for flight things like snacks - and count that as my personal item. My carry on bag goes overhead.
My husband uses his laptop bag for his.

It really depends on the airline.

Ryanair is one of the most restrictive:
https://www.ryanair.com/gb/en/plan-trip/flying-with-us/baggage-policy

Most Canadian/American airlines are less restrictive. This is from Delta:

Approved Personal Items
1 purse, briefcase, camera bag or diaper bag
or 1 laptop computer (computers cannot be checked)
or 1 item of a similar or smaller size to those listed above

Best to check with your airline so you don't wind up with an unexpected fee.

We are going carry-on only in a few weeks. Two carry on pieces and two personal items. For us the personal items are a Pacsafe shoulder bag and a smaller backpack.

It totally depends on the airline. Only some airlines allow two carry-on items.
As well as Ryanair, Easyjet has a strict "one item carry-on" policy. See here: http://www.easyjet.com/en/help/baggage/cabin-bag-and-hold-luggage

I suggest you check the relevant airline website.

In six trips we have taken a roller bag and a personal item. Usually a soft side day bag that fits under the seat. We place all of our seat items, wipes, ear phones, snacks, small flashlight on the over night flights, etc in a separate small bag we can easily remove and more often than not we place the bigger bag in the overhead. Never been told we can’t. Short flights we do what ever we have to, often we are required to check them so the large personal bag goes with us. Buy one with a trolly sleeve for those trips through airport security lines and customs
Safe travels
Don
PM me and I will send you photos
PS we mostly fly Seattle direct the major cities in Europe. 9 hours. Delta, virgin Atlantic, KLM or British Air

Rick Steves Euro-tote has worked for me on transAtlantic flights -- squishable, fits under the seat, trolley sleeve for the rolling bag.

For starters, a "personal item" should be able to fit under the seat. But as said above, each airline will have its own definition. On the less restrictive airlines and tickets, just about anything you can fit under your seat will be OK. On more restrictive airlines, or if you've got one of those "Basic Economy" tickets with more carry-on restrictions, they may weigh and/or measure your personal item, to make sure it meets their particular specs.

I use Rick's Euro Flight Bag and it works well. My only problem is that, since it's a "cube" shape, it's not comfortable to carry for long distances (I put it over my suitcase handle), or to use at my destination (I have another bag for carrying around my water bottle, phone charger, umbrella, guidebook, etc).