-chan - Used between friends or with someone younger than you. Generally a very friendly suffix such as "Shippo-chan" as Kagome calls Shippo. Used between boyfriends and girlfriends.
-kun - Generally used for males, but can be used for females too. Someone who's close to you as a friend. Kagome calls Kouga "Kouga-kun" to show how she's friendly with him.
-san - Polite and formal. Used for someone at the same position as you or higher. You would use this with strangers if you didn't know them well. It basically means "Ms, Miss, Mr., or Mrs." Miroku calls Kagome "Kagome-san" out of respect.
-dono - Not frequently used in this anime, but in others, like Rurouni Kenshin. Indicates respect and is very polite.
-sama - Very high respect. Used when speaking about gods or someone like an emperor or king. In this case, Kikyo is called "Kikyo-sama" by the villagers that she lived with.
-sensei - Originally means "born earlier than me". Usually used towards your master, teacher, or someone wise in the literature and art. You would call your school teacher by this suffix.
onee-chan/san/sama - what you would call your sister. Souta calls Kagome "Kagome onee-chan". Kaede calls Kikyou "Kikyou onee-sama".
onii-chan/san/sama - what you would call your brother. Souta calls Inuyasha Inu-no-oniichan or the "Dog-eared brother".
jii-chan/san/sama - grandfather. Kagome calls her grandfather "Jii-chan".