This thesis describes the theory and applications of weak gravitational galaxy-galaxy lensing. In particular, constraining the galaxies' mean halo mass M200, concentration parameter c and ellipticity of the dark matter halo distribution ε1l. Weak gravitational lensing is the distortion of background galaxy images by matter distributions along the line of sight and is a unique astronomical probe of dark matter and consequently, dark energy on large scales.In order to better understand weak lensing theory this thesis begins with descriptions of the cosmological model, necessary distance definitions and a discussion on the observational evidence for dark matter and dark energy. Lensing theory, which introduces the tangential and cross components of the shear, is then described in detail. Subsequently we describe NFW density profiles and the shear induced by their matter distributions. A method for modelling NFW profiles is then presented due to its relevance for fitting models to the DES data.Currently weak gravitation lensing is limited by systematic errors that need to be corrected for. These systematics are discussed, followed by an analysis of the DES shear catalog used throughout this thesis. We found that over the whole catalog there is a non-zero mean component of the ellipticity ⟨ε1⟩.Galaxy-galaxy lensing is the distortion of background galaxy images induced by a foreground galaxy lens. The mean halo mass M200 and concentration parameter c completely define the dark matter halo profile of these foreground lens galaxies. We describe a model fitting method and apply it to find the lens LRGs have mean halo mass M200 = 4.81 ± 0.26 × 10^12 h^-1M⊙ and concentration parameter c = 9.80 ± 0.08 for the green catalog. The yellow catalog was found to have mean halo mass M200 = 4.77 ± 0.16 × 10^12 h^-1M⊙ and concentration parameter c = 9.81 ± 0.05. The 68 % confidence limit is conveyed in all the numbers.The impact of the number density of galaxies and the non-zero mean component of the ellipticity ⟨ε1⟩ on constraints for the foreground lens galaxies' ellipticity are investigated. Predictions indicate that although the current DES number density of galaxies is adequate, the non-zero mean component of the ellipticity ⟨ε1⟩ ∼ 10^-3 must be reduced to ∼ 10^-5 in order to accurately constrain the lens ellipticity.