Why does a t-shirt stretch when there is no elastane in it?

The stretch is in the structure, not the fiber. Each knit loop can deform and borrow yarn from its neighbors, then spring back. A woven cotton of identical fiber barely stretches because its threads are held straight and locked at every crossing.

Why do knits not need ironing?

A crease is a sharp fold held by the structure. Knit loops are springy and round, so they absorb the fold and relax back; the flat interlacements of a woven cloth take and keep the crease instead.

Side by side

Knit vs Woven: Jersey and Poplin

The deepest divide in fabric is not fiber but structure: knit or woven. Jersey and poplin make the cleanest pair, the t-shirt against the dress shirt. Jersey is one thread looped through itself, so it stretches, drapes, and never needs ironing. Poplin is warp crossing weft, so it is crisp, stable, and holds a pressed shape. Every garment in your wardrobe sits on one side of this line.

Jersey

No. 067

weft knit, stockinette · first documented Middle Ages

Poplin

No. 029

plain weave, fine crosswise rib · first documented 1400s

The differences

Aspect	Jersey	Poplin
Structure	Knit: a single yarn looped row into row (stockinette Vs).	Woven: warp and weft crossing at right angles.
Stretch	Inherent, even with no elastane: the loops give.	Almost none on the straight grain; only the bias gives.
Wrinkles	Barely wrinkles; loops absorb the fold.	Creases sharply; that is what ironing is for.
Edges and sewing	Curls at the edges and can ladder like a stocking.	Frays at the edges; otherwise stable to cut and sew.
The garment line	T-shirts, underwear, athleisure, anything pulled on.	Shirts, blouses, anything buttoned and pressed.

Which to choose

Knit when the garment must move and be lived in; woven when it must hold a shape and a press. The t-shirt and the dress shirt are not different qualities of one thing; they are different machines of cloth.

Common questions

Why does a t-shirt stretch when there is no elastane in it?: The stretch is in the structure, not the fiber. Each knit loop can deform and borrow yarn from its neighbors, then spring back. A woven cotton of identical fiber barely stretches because its threads are held straight and locked at every crossing.
Why do knits not need ironing?: A crease is a sharp fold held by the structure. Knit loops are springy and round, so they absorb the fold and relax back; the flat interlacements of a woven cloth take and keep the crease instead.

Full entry: Jersey Full entry: Poplin

Sources & References

1.Jersey (fabric), Wikipedia
2.Knitting, Wikipedia
3.Poplin, Wikipedia
4.poplin, Online Etymology Dictionary