Why does it matter that paper is acidic?
Paper is used for a wide variety of media documentation, exposed to countless chemical formulations of ink, depending on if it's used in a book, a document printed in an office, a photograph, etc. All of these inks will potentially impair the longevity of the media, as will environmental factors like temperature, humidity, bookworms, book lice, and the oils and residue transferred to the paper by contact with human hands.
Museums, libraries, national archives, and similar public media organisations, all strive to preserve historically, culturally, and artistically significant media for posterity. It's bad enough when the chemical composition of paper leaves a single document more vulnerable to damage from handling and its environment... but it is far worse when one document can cause damage to another document, by its mere proximity in the same enclosed space. This is by far the greatest threat presented by acidic paper in an archival environment: books breathe , volatile compounds evaporating off them into the air over time and interfering with other books nearby. These compounds make paper increasingly yellow, brittle, flammable, and susceptible to book parasites which locate books by detecting this odor.
Archival paper can last hundreds of years; some has lasted over a thousand years. Acidic paper - and archival paper exposed to acidic vapours - typically degrades within a single human lifespan.
Many paper producers market a wide variety of paper which they list as "archival quality," but these labels, on their own, are not to be trusted, as there is no formal regulatory standard for paper acidity, and most retailers do not test paper for acid before selling it. Neither do they store it sufficiently far from acidic paper, in a suitably climate-controlled environment: even if paper is truly acid-free, as its packaging states, it can become inadequate for an archive based on how the retailer stores and transports it.
Being nominally acid-free is also not enough, even with optimal storage. Paper needs to be free of lignin, a by-product of the paper-making process. Lignin undergoes slow chemical decomposition which emits acids, turning non-acidic paper acidic.. Paper that is acid-free today will become acidic over time if it contains lignin, and documents using this paper will yellow, fox, and become brittle and tacky (pages sticking together and crumbling when pulled apart), rendering the document unusable by future readers, and certainly unsuitable for any formal archive.
Okay, what about pH testing pens?
It is true that there are felt-tip pens which can be used on paper to detect if it is acidic right now, but these pens will not detect the presence of lignin, and a number of brands of testing pens have been shown to not work at all, or work only inconsistently.
If labels aren't reliable, and pH testing pens don't work half the time, then how can I tell if paper is acidic or not?
The current most reliable method for testing the current and future acidity of paper - as well as plastics, adhesives, and inks - is the Photographic Activity Test, which involves subjecting samples of the tested material to high temperatures and high humidity. The PAT is an ISO standard test, and it is described in detail in ISO 18916:2007 (E). It was developed by the Image Permanence Institute of the United States to test the quality of photographic storage materials, as photographs are even more sensitive to acidity than text documents. The IPI allows archives to submit material samples for testing, but the average hobbyist will find this financially prohibitive.
Well, that's no good. What can I do, at home, to make sure my documents last as long as possible?
If you're a hobbyist, not somebody working in a national archive, then the long-term relevance and value of anything you put to paper is largely going to be what value you assign to it personally. Maybe your kids or grandkids will see something worthwhile in your old journals, especially if you used some gorgeous calligraphy to compose them. Regardless, the natural wear and tear you inflict on your writing materials is already going to impair its longevity, and that means you don't have much to lose from lignin and acid vapour, that skin oil and humidity haven't done already. Rhodia brand paper is the gold standard for calligraphers and fountain pen enthusiasts at the time of this writeup, so that's where I'd recommend you look for long-lasting paper that plays nicely with a variety of inks and writing instruments. It has the lowest lignin content of any widely-available writing paper. Leuchtturm1917 paper is another suitable alternative, and if you treat it well when you're done writing on it, it will probably survive long enough for your descendants to appreciate what a fine armchair philosopher you were, back in your day.
EDIT: As Clockmaker has pointed out, cotton-linen paper contains little or no lignin, and can generally be trusted to be acid-free. It does not play nicely with many inks, however, so it is recommended you test it with whatever media you are creating, before committing to it.
Iron Noder 2016, 6/30