Paper comes from trees! Trees can be cut down just for paper, but some paper comes from trees cut down for building materials or furniture. Whole trees or the smaller pieces that can’t be used in other industries are chopped into smaller logs and transported to factories. In order to make pure paper, they’re then stripped of their bark and cleaned. The wood fiber sources required for pulping are 45% sawmill residue, 21% logs and chips, and 34% recycled paper. Around 10% of the wood used for making paper is taken from old-growth forests, which results in habitat loss for the living things in the area. More trees are cut down in sustainable ways, but laws vary in different counties and there are still trees that are removed by clear-cutting or removing all the trees in an area.

Clear cutting is an example of a non-sustainable way to harvest timber for furniture, building, and paper.
By Calibas - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=14518481
Once the bark is removed, they undergo a process called chipping, which chops the paper up into smaller chips.

Wood chips after being cleaned and before being turned into pulp.
Next, the paper is smashed into a pulp and broken down using both machines and chemicals. If the paper has been chemically pulped, often most of the colors have been removed. The pulp is then cooked, to remove lignin, which is a chemical that provides much of the color of wood. The pulp is then cleaned and sent off for papermaking. Older paper methods used to use chemicals that were harmful to the environment, but newer technologies allow us to re-use almost all chemicals at this step of the process.

Wet paper pulp after being processed from wood chips.

The microscopic structure of paper pulp.
PapermakingThe pulp is sprayed onto a moving wire mesh, which is then pressed into a thin mat that looks much more like the paper we use. The paper is finally dried using huge cast iron cylinders and rolled into huge rollers to be sent to finishing plants.

The microscopic structure of a sheet of paper.
FinishingThe final step in papermaking involves finishing the paper to provide the texture and quality that we need for different applications. Many different machines are used to smooth the paper, apply a gloss, or shiny finish for things like photo paper, and cut the paper into various sizes that can be sold.
RecyclingPaper is a material that can be recycled a few times before it loses its durability. The United States Environmental Protection Agency has found that recycling causes 35% less water pollution and 74% less air pollution than making virgin paper. Most paper from consumers or offices can be sent to a deinking mill. These mills will bleach the paper with chemicals to remove the ink and this paper can be made back into pulp again to go through the papermaking process. Recycling paper costs more and takes more time than making new paper from wood, but the process helps to save the environment from some of the pollution that's a result of the papermaking process.
References:Sixta, Herbert (2006). "Preface". Handbook of Pulp. 1. Wiley-VCH Verlag & Co KGaA. p. XXIII. ISBN 3-527-30999-3.
"Recycle on the Go: Basic Information". US Environmental Protection Agency. October 18, 2007.