Please explain banishment in "The Seafarer", "The Wanderer", and "Wife's Lament"?
All three of these poems are Anglo-Saxon elegies, meaning that they are mourning for something that they have lost.
In The Seafarer, the sailor is mourning the fact that he is becoming too old to do the work he loves which is sailing and living on the ships. He tells us that life on land is more comfortable, but that he does not relish the women, food, or drink. Even in the coldest, most uncomfortable weather, he simply loves being on the sea and his soul longs for it when he is not there. The banishment in this poem is the banishment of age...making him physically unable to further the work he enjoys.
In The Wanderer, the speaker is lamenting the loss of his lord, probably through death, and is suffering a "banishment" as his services are no longer needed on the property. He tells of the hardships of a life of transience and longs for a more settled life as he had before.
In The Wife's Lament, the speaker is lamenting the loss of her husband or lover and his family. It makes more sense being simply a "lover" since there is not legal connection, but in any sense, his death has caused her "banishment" from the family household and properties. Her pain of the separartion is apparent, and understandable since human beings are social beings.
In all of these poems, there is a hint at the speaker's former situtation as "heaven" and perfect, and the separation is symbolic of a separation from God and that perfection.