  Euclid is poised to make significant advances in the study of nearby galaxies
in the local Universe. Here we present a first look at 6 galaxies observed for
the Nearby Galaxy Showcase as part of the Euclid Early Release Observations
acquired between August and November, 2023. These targets, 3 dwarf galaxies
(HolmbergII, IC10, NGC6822) and 3 spirals (IC342, NGC2403, NGC6744), range in
distance from about 0.5 Mpc to 8.8 Mpc. Our assessment of the surface
brightness depths in the stacked Euclid images confirms previous estimates in
100 arcsec^2 regions of 1sigma=30.5 mag/arcsec^2 for VIS, but slightly deeper
than previous estimates for NISP with 1sigma=29.2-29.4 mag/arcsec^2. By
combining Euclid HE, YE, and IE into RGB images, we illustrate the large
field-of-view covered by a single Reference Observing Sequence, together with
exquisite detail on parsec scales in these nearby galaxies. Radial surface
brightness and color profiles demonstrate galaxy colors in agreement with
stellar population synthesis models. Standard stellar photometry selection
techniques find approximately 1.3 million stars across the 6 galaxy fields.
Euclid's resolved stellar photometry allows us to constrain the star-formation
histories of these galaxies, by disentangling the distributions of young stars,
as well as asymptotic giant branch and red giant branch stellar populations. We
finally examine 2 galaxies individually for surrounding satellite systems. Our
analysis of the ensemble of dwarf satellites around NGC6744 reveals a new
galaxy, EDwC1, a nucleated dwarf spheroidal at the end of a spiral arm. Our new
census of the globular clusters around NGC2403 yields 9 new star-cluster
candidates, 8 of which with colors indicative of evolved stellar populations.
In summary, our investigation of the 6 Showcase galaxies demonstrates that
Euclid is a powerful probe of the anatomy of nearby galaxies [abridged].
