What meter is this?

While "Peculiar Meter" or "Particular Meter" are the commonest uses of P.M. in hymnals and tunebooks, sometimes it means "Psalm Meter" (8,7.8.7); note also the definitions below of "L.P.M., C.P.M., and S.P.M." quoted from https://hymnody.askdefine.com/

• H.M. - Hallelujah Meter; a six-line stanza of which the first four lines are trimeter and the last two are tetrameter, which rhymes most often in the second and fourth lines and the fifth and sixth lines (6/6/6/6/8/8).
• L.P.M. - Long Particular Meter; a six-line stanza of iambic tetrameter (8/8/8/8/8/8).
• M.T. (or 12s.) - Meter Twelves; a quatrain in anapestic hexameter (12/12/12/12).
• C.P.M. - Common Particular Meter; a six-line stanza of which the first, second, fourth and fifth lines are iambic tetrameter, and the third and sixth lines are iambic trimeter (8/8/6/8/8/6).
• P.M. - may stand for Psalm Meter (more commonly known as 8s.7s), Particular Meter, or Peculiar Meter (each indicating poetry with its own peculiar, non-standard, meter).
• S.P.M. - Short Particular Meter; a six-line stanza of which the first, second, fourth and fifth lines are iambic trimeter, and the third and sixth lines are iambic tetrameter (6/6/8/6/6/8).

Also bear in mind that some editors call anything that doesn't fall into one of the commoner metrical categories "P.M." and/or "Irregular", others distinguish between these terms, using P.M. for lyrics whose meters are definite, regular, but rare, as opposed to "Irregular" which denotes lyrics whose syllable counts do not bear definition, such as a six-line text of 5.8.7.7.3.6 syllables, or lyrics whose stanzas (usually through the use or omission of ties, slurs, etc.) vary in their syllable counts. A good example of the latter is "The First Nowell" aka "The First Noel", where the first two lines of the first stanza are 9 and 12 syllables respectively, whereas the syllable count for the same two lines of the second stanza is 8 and 9. 

Hope this helps. As far as the meaning of P.M. in the particular hymnal you're talking about, you'd have to look at some of the P.M. texts and see if they share a common syllabic line structure or if they vary all over the map. If the hymnal suggests tune names for any of these, you might look them up here or in other hymnals and see what others say. And even if tunes are not suggested, you could look up the first lines here and see what the consensus is in the hymnals that contain that hymn.