Gamma-ray burst observations provide a great opportunity for cosmography in high redshift. Some tight correlations between different physical properties of GRBs are discovered and used for cosmography. However, data selection, assumptions, systematic uncertainty and some other issues affect most of them. Most importantly, until the physical origin of a relation is understood, one should be cautious to employ the relation to utilize Gamma ray bursts for cosmography. In the first part of this dissertation, I use Liang-Zhang correlation to constrain ¦« Cold Dark Matter standard cosmology and a particular class of brane cosmology (brane-induced gravity model). With the most probable model being ¦¸_m=0.23 and ¦¸_¦«=0.77 for flat ¦«CDM cosmology and ¦¸_m=0.18 and ¦¸_(r_c )=0.17 for flat brane-induced gravity cosmology, my result for the energy components of these two models is comparable with the result from SNIa observation. With average uncertainty of distance modulus being 0.2771, the two discussed cosmologies are indistinguishable using my current sample of GRB with redshift ranging between 0.1685 and 3.2. I argue that by expanding my sample and adding more low and high redshift GRBs and also with improvement in using GRB for cosmography, we might be able to distinguish between different cosmological models and tighten the most probable model. Looking into correlation and evolution of GRB prompt emission and afterglow has many advantages. It helps to open windows to comprehend the physics of GRBs and examine different GRB models. It is also possible to use GRB correlation as an accurate redshift estimator and more importantly to constrain the cosmological parameters. XRT flares of GRB afterglow are thought to be the result of central engine activity. Studying this component leads us to understand GRB flare and central engine nature. In the next part of this dissertation, I study the correlation and evolution of different prompt emission and afterglow GRB properties and some GRB flare-based quantities. Considering instrument bias and selection effect, I conclude some well-correlated correlations and establish some property evolution. The correlation between average luminosity and isotropic ¦Ã-ray energy, energy of plateau and isotropic ¦Ã-ray energy and luminosity at break time and break time and evolution of plateau energy are well established. It is also realized that the apparent evolution of isotropic ¦Ã-ray energy and average luminosity is due to the instrumental flux threshold. With expanding the sample of GRB and accommodating more GRBs with XRT flares to my sample, I can reevaluate my result more firmly and confirm or rule out some hard to assert results due to limited number of data. In search for physically motivated GRB relation, analyzing the thermal component of GRB prompt emission, I derive two well-correlated relations. They are between calculated and estimated flux of the GRB thermal component for the co-moving bolometric and co-moving detector band-pass range of spectrum. In this study, three samples of Swift, pre-Swift and combined samples are used. The quality of this correlation is comparable with the Ghirlanda relation in terms of Spearman rank correlation parameters (correlation coefficient and correlation significance) and reduced ¦Ö^2of best fit. These results for the Swift GRB sample for co-moving bolometric range of spectrum are 0.81, 4.07¡Á¡¼10¡½^(-7) and 0.66 respectively. The derived correlations also imply a E_(¦Ã,iso)-E_peak^4 relation that provides physical insight to E_¦Ã-E_peak Ghirlanda correlation. Three scaling coefficients are employed to study these correlations. Monte Carlo statistics indicates that the existing correlations are independent of these constants. For Swift and combined sample 73% - 84.8% successes are recorded. Therefore, it is expected by determining these constants, the tightness of these correlations will further improve.