Aunt Jennifer's tigers prance across a screen, (1)
The poem begins in a moment of strength. The tigers belong to Aunt J, and they actively prance across her tapestry.
Aunt Jennifer's finger fluttering through her woolFind even the ivory needle hard to pull. (5-6)
Here, Aunt J seems like the opposite of her tigers. It's hard for her to do her needlework, which is not exactly a difficult feat! She's also "fluttering," which seems weak compared to her brave, chivalric tigers.
The massive weight of Uncle's wedding bandSits heavily upon Aunt Jennifer's hand. (7-8)
Here we find out why it's hard for her to do her needlework. Aunt J is being held down by the (symbolic) weight of "Uncle's wedding band." She's wearing the band, but it's not even hers—it's "Uncle's." Aunt Jennifer is controlled and held back from her life by this "massive weight"—by the weight of her marriage. In short: bad times.