We present the measurements and cosmological implications of the galaxy two-point clustering using over 4.7 million unique galaxy and quasar redshifts in the range 0.1 < z < 2.1 divided into six redshift bins over a ∼ 7,500 square degree footprint, from the first year of observations with the Dark Energy Spectroscopic Instrument (DESI Data Release 1). By fitting the full power spectrum, we extend previous DESI DR1 baryon acoustic oscillation (BAO) measurements to include redshift-space distortions and signals from the matter-radiation equality scale. For the first time, this Full-Shape analysis is blinded at the catalogue-level to avoid confirmation bias and the systematic errors are accounted for at the two-point clustering level, which automatically propagates them into any cosmological parameter. When analysing the data in terms of compressed model-agnostic variables, we obtain a combined precision of 4.7% on the amplitude of the redshift space distortion (RSD) signal reaching a similar precision with just one year of DESI data than with twenty years of observation from the previous generation survey. We also analyse the data to directly constrain the cosmological parameters within the ΛCDM model using perturbation theory and combine this information with the reconstructed DESI DR1 galaxy BAO. Using a Big Bang Nucleosynthesis Gaussian prior on the baryon density parameter, ωb , and a weak Gaussian prior on the spectral index, ns , we constrain the matter density is Ω m = 0.296±0.010 and the Hubble constant H 0 = (68.63 ± 0.79)[km s-1Mpc-1]. Additionally, we measure the amplitude of clustering σ 8 = 0.841±0.034. The DESI DR1 galaxy clustering results are in agreement with the ΛCDM model based on general relativity with parameters consistent with those from Planck. The cosmological interpretation of these results in combination with DESI DR1 Ly-α forest data and external datasets are presented in the companion paper [1].