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Pilcrow is the paragraph symbol (¶), and octothorp is the hash mark or number sign symbol (#) on touch-tone telephone keypads. I knew octothorp from my years working in the telephone business, but I just learned pilcrow from Michael Quinion, World Wide Words, Issue 586 (May 3, 2008, also available here):

The word is delightful, not least because it gives no clue at all to what it means or where it might come from. The recently revised entry for it in the Oxford English Dictionary says that it is "now chiefly historic", which I rather dispute, since it's easy to find examples in current books on typography and it continues to be used in standards documents that list character sets.

What makes it truly weird is that the experts are sure it's a much bashed-about transformation of paragraph. This can be traced back to ancient Greek paragraphos, a short stroke that marked a break in sense (from para-, beside, plus graphein, write). The changes began with people amending the first r to l (it appeared in Old French in the thirteenth century as pelagraphe and pelagreffe). Then folk etymologists got at it, altering the first part to pill and the second to craft and then to crow. The earliest recorded version was pylcrafte, in 1440; over the next century it settled down to its modern form.

The paragraph symbol, by the way, isn't a reversed P as you might guess. It's actually a script C that was crossed by one or two vertical lines. The letter stood for Latin capitulum, chapter.

On the Greek paragraph symbol, see William A. Johnson, "The Function of the Paragraphus in Greek Literary Prose Texts," Zeitschrift für Papyrologie und Epigraphik 100 (1994) 65-68.

Walter W. Skeat, Notes on English Etymology: Chiefly Reprinted from the Transactions of the Philological Society (Oxford: Clarendon Press, 1901), pp. 215-216, describes in detail the phonetic development of the word pilcrow:

First of all, the Lat. paragraphus became F. paragraphe. This is given by Cotgrave, who has: 'paragraphe, a Paragraffe, or Pillcrow; ... as much as is comprehended in one sentence or section.' The next form is paragraffe, just cited as an E. word from Cotgrave. After this, the middle a was dropped, and an excrescent t added at the end. This is quoted by Way from the Ortus Vocabulorum: 'Paragraphus, Anglice, a pargrafte in writing.' The next step is the corruption from pargrafte to the form pylcrafte in the Promptorium. This is rather violent, but we must remember that the change of r to l is the commonest of all changes in every Aryan language, that the prefixes par- and per- were convertible, and that the change from per- to pil- occurs in the common English word pilgrim, in which per- passes into pil- through the Ital. pell- in pellegrino. This shows the precise process; pargrafte became *pergrafte, then *pelgrafte, then *pilgrafte, and finally pilcrafte, with c for g. The change from g to c easily took place when the original form had become entirely obscured. After this, a further corruption took place, from pilcrafte to pilcrow. This was due to mere laziness. The excrescent t was again dropped, giving pilcraf, and then the -craf became -crow. Hence we get the full order of successive forms, viz. paragraphe, paragraffe, *pargraf, pargrafte, *pergrafte, *pelgrafte, *pilgrafte, pilcrafte, *pilcraf, pilcrow. Not all of these forms are found, but a sufficient number of them appear to enable us to trace the complete process; at the same time, it is highly probable that some of these steps were passed over by a sudden leap. We may assume, as sufficiently proved, that pilcrow and paragraph, words used with precisely the same meaning, are mere doublets.

Oxford English Dictionary s.v. octothorp:

Forms: 19- octothorp, 19- octothorpe. [Origin uncertain; perh. < OCTO- comb. form + the surname Thorpe (cf. THORP n.: see note below).

The term was reportedly coined in the early 1960s by Don Macpherson, an employee of Bell Laboratories:

1996 Telecom Heritage No. 28. 53 His thought process was as follows: There are eight points on the symbol so octo should be part of the name. We need a few more letters or another syllable to make a noun... (Don Macpherson..was active in a group that was trying to get Jim Thorpe's Olympic medals returned from Sweden). The phrase thorpe would be unique.

For an alternative explanation see quot. 1996; in a variant of this explanation, the word is explained as arising from the use of the symbol in cartography to represent a village.]

The hash sign (#), as it appears on the buttons of touch-tone telephones and some other keypads.

1974 Telephony 25 Feb. 16/1 A few months ago, a story traveled through the Bell System that the familiar symbol '#'..at long last had a name: 'octothorp'. 1975 Vancouver Province 15 Nov. (Canad. Mag.) 32 Punch an octothorpe when you reach your desk every morning, and the accounting department automatically registers you in. 1987 Radio & Electronics World Feb. 47/1 As well as the numbers 1 to 9 and 0, you also have buttons marked with a star and square (also known as hash or octothorp). 1996 New Scientist 30 Mar. 54/3 The term 'octothorp(e)' (which MWCD10 dates 1971) was invented for '#', allegedly by Bell Labs engineers when touch-tone telephones were introduced in the mid-1960s. 'Octo-' means eight, and 'thorp' was an Old English word for village: apparently the sign was playfully construed as eight fields surrounding a village.

But for a different explanation and spelling see Douglas A. Kerr, The ASCII Character "Octatherp". Kerr was responsible for the selection of the non-numeric symbols * and # for use on the touch-tone telephone keypad. He attributes the invention of the word to John C. Schaak and Herbert T. Uthlaut:

They told me that they had read with interest the part of my report in which I regretted the absence of a unique typographical name for the character "#", and said they had solved my problem by coining one, "octatherp". They said that it had no etymological basis, but they had been guided by one principle. They said they were irritated that I had rejected some candidate characters they thought were good on the basis of lack of compatibility with emerging international standards (with which the Bell System had a tradition at the time of little interest). Thus, they said, as a way of getting even, they had included in the name the diphthong "th", which of course does not appear in German and several other languages and thus might be difficult for users of those languages to pronounce, which would serve them right.

Kerr started using the word in field memoranda, and he relates:

Before long, we were seeing, in non-Bell System publications, similar notes about the octatherp, sometimes accompanied by fanciful (and of course completely bogus) etymological explanations, such as "the prefix 'octa' refers to the eight tips of the four strokes of the character".

One author opined that "therp" was obtained by corruption of the German word "dorf", meaning village. He said he was not exactly sure of the logical trail there.