How many luggage are allowed in international flights and what should be their maximum weights and dimensions?

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Here is a brief glossary and quick explanation of the things that might surprise you in Finnair's description:

Carry-on is what you take with you in the cabin. It should generally fit in the overhead bins or under the seats and not lie around, especially during take-off and landing. Because of this, there is a separate maximum length, width and height (which can typically be checked at once by trying to fit the bag in a metal box - beware: wheels and handles count!). On most airlines, there is also an explicit weight limit (sometimes 7 or 10 kg) but some low-cost airlines do not limit weight per se and only require that you should be able to lift your luggage by yourself.
What you surrender at check-in and get back at your destination is usually called “hold luggage” (because it goes into the… hold – the word is also used when talking about ships). It differs depending on the airlines, fare, flight distance and sometimes even by destination but one or two pieces of 23 kg is a rather typical allowance for this.
When there is only one “dimension” like 158 cm, it means that the sum of length + height + width must be under 158 cm. You're free to take a long, thin or flat piece of luggage, a typical suitcase or a perfect cube, the shape does not matter. This typically applies to hold luggage because the shape of carry-on luggage is restricted by the size of the overhead bins.

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Source: stackoverflow.com