What is this "Table of astronomy" about? I found this picture while looking for some tattoo ideas and would like to know what these sketches are supposed to mean.   

Im especially interested in figures 1, 21, 53 and 56.  

It appears to be a scan of page 164 of the Cyclopædia (an Universal Dictionary of Arts and Sciences), from 1728. Higher Resolution
Fig. 1 and many others on that page seems to be diagrams of various trigonometric relations, used e.g. to convert between coordinates of celestial objects. But I'm not sure about this particular one.

Fig. 21 is a armillary sphere, i.e. a physical model showing the position of objects in the sky. Because this particular model has Earth in its center, it's also called a Ptolemaic sphere. Had it been centered on the Sun, it would have been a Copernican sphere, seen in Fig. 22.

Fig. 53 is a quadrant, which is  an instrument used to measure angles, e.g. between stars, or longitude and latitude.

Fig. 56 is a horary quadrant, which is used for finding the time of the day, using the Sun. This particular version was made of wood, paper, and brass by Henry Sutton around 1658.



Additionally,

Fig. 2 shows a diagram of the diurnal motion, i.e. the apparent motion of celestial objects around Earth.

Fig. 3 is titled "Aspect Trine Tetragon". Aspect is the astrological term for the angle between planets. This diagram shows a triangle ("Trine"), a square ("Tetragon"), and a pentagon.

Figs. 11 and 12 are micrometers (not to be confused with the length unit $\mu\mathrm{m}$), which is a device used to measure lengths very accurately. It was invented by the British astronomer William Gascoigne.

Figs. 13–20 are diagrams of the phases of the Moon and drawings of its surface. It seems the person who drew these diagrams accidentally swapped "1" and "8" in Fig. 18, since the figures run like 13, 14, 15, 16, 17, 81, 19, 20.

Fig. 34–38 illustrate how Lunar eclipses occur.

Figs. 43, 44, and 45 show the positions and orbits of the planets, Earth, and the Sun as depicted in the Ptolemaic (geocentric), Copernican (heliocentric), and Tychonic system, respectively. The latter used the mathematics of the Copernican system, but held Earth in its center since that's what Tycho Brahe believed.

Fig. 57 shows a gnomon, which is the part of a sundial casting the shadow.

Fig. 58 shows a globe. Fig. 59 seems to show how the spherical sky can be mapped onto a flat map by dividing it into 12 elongated stripes, each going from the south pole to the north pole, with a central width of 2 hours, or 30º. One of the stripes (around R.A. ~ 15 h, it seems) is shown in Fig. 60 with the constellations in that region.