The thesis deals with a plasma diagnostic device, the Hairpin Probe, popularly used for measuring electron density in rarefied gaseous plasma. Electron density, n_e, is an important plasma parameter as electrons are mainly responsible for inelastic collision with background neutrals resulting in ionization, excitation, and various chemical processes in plasma. Besides, the basic plasma parameters such as plasma frequency, Debye length, plasma permittivity, and plasma conductivity are all based on n_e. Therefore accurate measurement of n_e is fundamentally desirable for quantifying the state of plasma. The underlying principle relies on measuring the effective permittivity of medium surrounding the hairpin. If length of hairpin is chosen equal to a quarter-wavelength of an incident microwave signal, a standing wave is set-up along its length. Under this condition, a strong absorbance of incident em signal is observed as hairpin is driven to resonance. When hairpin is immersed in plasma, the cold plasma permittivity is related to ne. However if adjacent dielectrics are present in the vicinity of probe, it can adversely affects the measurement. As one of the practical applications of hairpin, high refractory material is coated on the probe surface when applied to reactive etch plasmas. However, the contribution of external dielectric on probe resonances in plasma is an outstanding problem. In this thesis, we have primarily addressed the above issues. A comprehensive study is also devoted towards application of probe in strongly magnetized plasmas. The electrons gyro motion modifies the plasma permittivity and results in the observance of dual resonances as compared with non-magnetized plasmas. The other important issues addressed are different loss mechanisms causing dispersion of resonance signal in plasma. This is particular topic of interest in order to broaden the range of n_e measurement by probe.