Airlines will always differentiate between "checked baggage" and "carry-on baggage/items."

CHECKED
A checked bag is what you refer to as 'proper baggage'; you check it in with the airline, generally at the airline counter, it is taken from you, and travels in the baggage compartment of the aircraft. I'm not an international traveler, so I can't comment on airlines outside of the US, but US carriers usually charge a fee for checked bags.

Some airlines let you bring one checked bag free and charge for a second, and one airline that I know of, Southwest, doesn't charge you for up to two checked bags (more than two, they charge for).

All airlines, as far as I know, place a weight limit on checked bags; if the bag is over the limit, it can still fly, but you will be charged an additional fee for the extra weight. On US airlines, the weight limit is usually 50 pounds (22.6kg).

CARRY-ONS
Carry-on bags and items are those which you carry onto the plane yourself and travel in the cabin with you. Policies on these items differ from airline to airline, but most of the US airlines allow you one carry-on bag, plus one smaller "personal item".

Carry-ons are limited not by weight, but by size; this is due to the limited space in the overhead and under-seat storage compartments. Each airline has their own size limits for carry-on bags, but they're generally in line with the common small rolling cases with extendable handles known as rollaboards.

Personal items would include such things as women's purses, laptop bags, attache cases, a child's diaper bag, or other miscelanous small objects, and these items size limits are a little more nebulous. A good rule of thumb, however, is that to qualify as a personal item, an item must fit in the underseat storage area without blocking the floor space used to get into and out of the seat row.

Most airlines don't charge for carry-ons, but a few have started, since fees for bags and other services have been the biggest money-maker in the air travel industry over the last 5 years or so.

If you are among the last to board a plane, and have carry-on items that won't fit under your seat, you might find that the overhead compartments are full before you board. If this is the case, this larger carry-on item might have to be gate-checked. I've never had it happen to me, but the way it's been explained to me, a gate-checked item is something that you try to bring into the cabin but there is no space for it, so it is given to the baggage handlers and stowed in the baggage compartment. However, gate-checked items don't get sent to Baggage Claim with the checked bags; gate-checked items are brought up form the baggage compartment when the plane lands, and are lined up in the jetbridge just outside of the plane's door, so you can retrieve them as you deplane.