  The Euclid ERO showcase Euclid's capabilities in advance of its main mission,
targeting 17 astronomical objects, from galaxy clusters, nearby galaxies,
globular clusters, to star-forming regions. A total of 24 hours observing time
was allocated in the early months of operation, engaging the scientific
community through an early public data release. We describe the development of
the ERO pipeline to create visually compelling images while simultaneously
meeting the scientific demands within months of launch, leveraging a pragmatic,
data-driven development strategy. The pipeline's key requirements are to
preserve the image quality and to provide flux calibration and photometry for
compact and extended sources. The pipeline's five pillars are: removal of
instrumental signatures; astrometric calibration; photometric calibration;
image stacking; and the production of science-ready catalogues for both the VIS
and NISP instruments. We report a PSF with a full width at half maximum of
0.16" in the optical and 0.49" in the three NIR bands. Our VIS mean absolute
flux calibration is accurate to about 1%, and 10% for NISP due to a limited
calibration set; both instruments have considerable colour terms. The median
depth is 25.3 and 23.2 AB mag with a SNR of 10 for galaxies, and 27.1 and 24.5
AB mag at an SNR of 5 for point sources for VIS and NISP, respectively.
Euclid's ability to observe diffuse emission is exceptional due to its extended
PSF nearly matching a pure diffraction halo, the best ever achieved by a
wide-field, high-resolution imaging telescope. Euclid offers unparalleled
capabilities for exploring the LSB Universe across all scales, also opening a
new observational window in the NIR. Median surface-brightness levels of 29.9
and 28.3 AB mag per square arcsec are achieved for VIS and NISP, respectively,
for detecting a 10 arcsec x 10 arcsec extended feature at the 1 sigma level.
