To write these is to write pieces of poetry or prose (or other texts) which try to do without specific letters.
Some choices might prove to be difficult: the letter E is often selected for disuse when trying to show some skill, since it is the most common letter in most lexicons. Ernest Vincent Wright did this in one of his novels in 1939, to show up those who thought it couldn't be done. It popped up once more in history when interest got renewed by Georges Perec with his 1969 French novel, supported by the group, OuLiPo. Other times, the decision is multiple letters should be done without, which forces even more funky sidestepping. Some choose to use only the letter E, resisting the other four: this begets prose like in Perec's 1972 novel, Les revenentes, or its English rewriting, The Exeter Text: Secrets, Jewels, Sex. But worry not, for vowels comprise only 5 of our 26 letters. To be more specific, T is the second most common English letter, while N is the fifth—their omission could provide the difficulty which one might seek in lipo-writing. Removing multiple letters is fun, too.
Writing these sorts of works requires you to settle into the right mindset. The trick is to notice which words must be dispensed with, finding fitting substitutes which don't seem very esoteric or uncommon. The issue of word overuse must be inspected, too; just think, how horrid it would be to seem obvious or repetitive when writing with such unique, interesting hurdles in your work! Some constructions require little thought in remedying for your missing letter—e.g. using 'Mr.' removes the E in 'Mister' surreptitiously—while others prescribe completely new sets of words, or even rethinking your choice of tense—no more verbs ending in '-ed'!
More concretely, consider this: 'the' is the most commonly used English word. You must do without, however, if you withhold T, H, or E. Some other often-used words now off limits, should you block out the letter E: 'he', 'she', 'me', 'be', 'we', 'they', 'there', 'one', 'three', 'seven', 'get', 'when'. The list is mostly pronouns, so you quickly figure out how to live without them, but still, the point is there: you will find quite the number of common words to dodge when writing without common letters! Likewise, with the letter I removed, you must not use 'it', 'in', 'will', 'with', 'his', 'think', or 'first', not to mention the big first person pronoun, 'I'. In short, it might be deceptively simple when you begin writing, so equip yourself with fresh, new synonyms often.
To those trying to "get into" this writing style: it might feel difficult, sometimes. Rightly so, but still, it will. Expect nerves to be tested, tortured even, should you come upon the one evil sentence, the one insipid thought, otherwise not of much consequence, but which must be expressed; it, however, simply refuses to work without your missing letter(s). You'll settle on some substitute, but it'll bug you endlessly. You might get inspired, plug up the hole with something truly perfect, but, more likely, you'll just need to move on. More words to write. Just don't give up. Not now, in the thick of it. It'll come in time.
You know, I wrote this node with some missing letters, myself. Proof of concept, etc. One of them, the letter Z, isn't honestly the biggest triumph one could perform. There is, however, another.