This answer intends merely to build on the nice ones already given and summarize some of their content, in order to give an idea of what will be found in some of the references before consulting them. My apologies if memory has faded the details following into serious errors.
The original question as asked, is answered already by Wirtinger, e.g. in his classic work Untersuchungen uber Thetafunctionen, on page 113, section 54, Uber die Thetafunctionen von zwei Variablen, lines -13,-14. He says (possibly very roughly):
“Each system of “eigentliche” (non trivial?) everywhere tangent conics to a plane quartic curve $C$ defines a Kummer surface, on which the curve $C$ lies. There are thus $63$ such Kummer surfaces.”
The point is that a plane section of a Kummer surface determines not just a genus three curve but a connected double cover of it, for which the Kummer surface is associated to the corresponding Prym variety, and a genus 3 curve has $2^{2g} -1 = 63$ non trivial double covers corresponding to the 63 non canonical square roots of twice the canonical class.
In general, there are (rough) correspondences between any two of the following data:
a curve $C$ of genus 3 equipped with a double cover;
a curve $C$ of genus 3 equipped with a half period;
a plane section $C$ of a Kummer surface;
a principally polarized abelian surface containing a genus 5 curve $C'$ representing twice the “minimal” class, i.e. a divisor in the system $|2\Theta|$.
a (semi stable, even) $\mathbb{P}^1$ bundle over a genus 2 curve $D$.
E.g. a "non trivial everywhere tangent conic" (not necessarily effective but not twice a line) to a plane quartic $C$ determines a half period on the genus 3 curve C;
a half period (on $\mathrm{Pic}(C)$) determines a double cover of a section (isomorphic to $C$) of the trivial line bundle on the curve $C$;
a plane quartic section $C$ of a Kummer surface inherits a double cover from that of the Kummer;
a double cover of a genus 3 curve $C$ determines as in rita's answer a Prym variety $P$ and a plane section isomorphic to C of the Kummer surface of $P$;
a double cover of a genus 3 curve $C$ induces a $\mathbb{P}^1$ bundle over the genus 2 theta divisor $D$ of the associated Prym variety via the Abel - Prym map;
a $\mathbb{P}^1$ bundle on a genus 2 curve $D$ yields a curve $C'$ in the system $|2\Theta|$ on the Jacobian of $D$, which parametrizes effective twists of an associated rank $2$ vector bundle.
These statements from memory will surely require correction by current experts, but may be useful as a sketch.
edit: Indeed perusing the paper of Verra seems to show that some of these data (such as #4) must be considered modulo the action of the group of points of order two on the abelian variety.
http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002329107&IDDOC=161897 (Alessandro Verra, "The fibre of the Prym map in genus three", Math. Ann. 276, no. 3, 1987)
In addition to the references above to Wirtinger, Verra, and Birkenhake-Lange, one may consult Narasimhan - Ramanan: Moduli of vector bundles on a compact Riemann surface, Annals of Math (1969), and perhaps:
http://msp.org/pjm/1999/188-2/pjm-v188-n2-p09-s.pdf (R. Smith, R. Varley, "On the geometry of two dimensional Prym varieties", Pacific Journal of Mathematics, volume 188, no. 2, 1999)
