【Dismiss】 , 【discharge】 , 【cashier】 , 【drop】 , 【sack】 , 【fire】 ,  【bounce】 are comparable when they mean to let go from one′s employ or service. 【Dismiss】 basically denotes a giving permission to go.

When used in respect to employment it carries apart from the context no suggestion of the reason for the act and is, therefore, often preferred as the softer or as the more comprehensive term.

【Discharge】 is usually a harsher term, implying dismissal for cause and little or no likelihood of being called back.

Only in military and court use does it, when unqualified, carry no implication of dissatisfaction on the part of the employer.

【Cashier】 implies a summary or ignominious  【discharge】 from a position of trust or from a position that is high in the scale.

【Drop】 , 【sack】 , 【fire】 , and  【bounce】 are all rather informal.

【Drop】 is a common and colorless synonym of 【dismiss】 .

【Sack】 stresses a being discarded or thrown out of employ while 【fire】 stresses a dismissal as sudden and peremptory as the action of firing a gun and 【bounce】 , a kicking out.