The Collaborative International Dictionary of English v.0.48:

Pillow lace \Pil"low lace\
   Lace made by hand with bobbins on a pillow.
   [Webster 1913 Suppl.]

The Collaborative International Dictionary of English v.0.48:

Pillow \Pil"low\, n. [OE. pilwe, AS. pyle, fr. L. pilvinus.]
   1. Anything used to support the head of a person when
      reposing; especially, a sack or case filled with feathers,
      down, hair, or other soft material.
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            [Resty sloth] finds the down pillow hard. --Shak.
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   2. (Mach.) A piece of metal or wood, forming a support to
      equalize pressure; a brass; a pillow block. [R.]
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   3. (Naut.) A block under the inner end of a bowsprit.
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   4. A kind of plain, coarse fustian.
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   Lace pillow, a cushion used in making hand-wrought lace.

   Pillow bier [OE. pilwebere; cf. LG. b["u]re a pillowcase],
      a pillowcase; pillow slip. [Obs.] --Chaucer.

   Pillow block (Mach.), a block, or standard, for supporting
      a journal, as of a shaft. It is usually bolted to the
      frame or foundation of a machine, and is often furnished
      with journal boxes, and a movable cover, or cap, for
      tightening the bearings by means of bolts; -- called also
      pillar block, or plumber block.

   Pillow lace, handmade lace wrought with bobbins upon a lace
      pillow.

   Pillow of a plow, a crosspiece of wood which serves to
      raise or lower the beam.

   Pillow sham, an ornamental covering laid over a pillow when
      not in use.

   Pillow slip, a pillowcase.
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The Collaborative International Dictionary of English v.0.48:

Lace \Lace\ (l[=a]s), n. [OE. las, OF. laz, F. lacs, dim. lacet,
   fr. L. laqueus noose, snare; prob. akin to lacere to entice.
   Cf. Delight, Elicit, Lasso, Latchet.]
   1. That which binds or holds, especially by being interwoven;
      a string, cord, or band, usually one passing through
      eyelet or other holes, and used in drawing and holding
      together parts of a garment, of a shoe, of a machine belt,
      etc.
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            His hat hung at his back down by a lace. --Chaucer.
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            For striving more, the more in laces strong
            Himself he tied.                      --Spenser.
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   2. A snare or gin, especially one made of interwoven cords; a
      net. [Obs.] --Fairfax.
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            Vulcanus had caught thee [Venus] in his lace.
                                                  --Chaucer.
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   3. A fabric of fine threads of linen, silk, cotton, etc.,
      often ornamented with figures; a delicate tissue of
      thread, much worn as an ornament of dress.
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            Our English dames are much given to the wearing of
            costly laces.                         --Bacon.
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   4. Spirits added to coffee or some other beverage. [Old
      Slang] --Addison.
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   Alen[,c]on lace, a kind of point lace, entirely of
      needlework, first made at Alen[,c]on in France, in the
      17th century. It is very durable and of great beauty and
      cost.

   Bone lace, Brussels lace, etc. See under Bone,
      Brussels, etc.

   Gold lace, or Silver lace, lace having warp threads of
      silk, or silk and cotton, and a weft of silk threads
      covered with gold (or silver), or with gilt.

   Lace leather, thin, oil-tanned leather suitable for cutting
      into lacings for machine belts.

   Lace lizard (Zool.), a large, aquatic, Australian lizard
      (Hydrosaurus giganteus), allied to the monitors.

   Lace paper, paper with an openwork design in imitation of
      lace.

   Lace piece (Shipbuilding), the main piece of timber which
      supports the beak or head projecting beyond the stem of a
      ship.

   Lace pillow, and Pillow lace. See under Pillow.
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WordNet (r) 3.0 (2006):

pillow lace
    n 1: a handmade lace worked on a pillow with threads wound on
         bobbins; the pattern is marked out on the pillow by pins
         [syn: pillow lace, bobbin lace]