This is kind of a big question, so I'll touch on certain aspects.

Cotton. Some of the earliest technological advances in the Industrial Revolution dealt with textiles; one famous innovation was the cotton gin, which processed cotton much more quickly than hand-workers could. This led to more cotton being planted, and much more cheap cotton being made. At the same time, you see the introduction of roller-printing, which quickly supplanted block printing - the former uses engraved rollers that can transfer a design to fabric continuously, while the latter uses an engraved block that must be picked up and applied to the fabric over and over again. The end result was that the previous cheap dress fabrics, linen and cotton with a very basic print, were replaced with much more complexly printed cottons.

Corsets. Stays were fairly commonly worn during the 18th century (depending on the country - France for example, saw fewer poor women in them than England), but until about the middle of the 19th century, foundation garments were made individually by professional corset-makers or at home. Ready-made corsets produced in factories (as well as the rest of the emerging ready-made garment industry) came about as a result of the development of the sewing machine, and advances in processing baleen (whalebone), then the innovation of steel boning. Earlier corsets could be exquisite but were often mainly functional; in contrast, later 19th century and early 20th century ones were made with aesthetics in mind very frequently. Once they became easily available at many price points, just about all women wore them.

Many of the same processes also led to the hoop skirt and bustle becoming first possible, then affordable and popular. These were made from the same steel used in watch springs, with a patented covering that was woven over them.

The industrial revolution gradually made textiles and clothing much less expensive, by mechanizing the processes by which they were produced.

This started with carding machines, progressed to the spinning jenny, went on to include mechanised looms, the cotton gin, sewing machines, and eventually laser cutters to cut patterns.

Originally, cotton (and wool and linen) were carded (combed) by hand, spun by hand, woven by hand, cut into patterns by hand, stitched into clothing by hand. (Also knit by hand, if knit rather than sewn.)

In 1748, Daniel Bourn in England invented the cylinder carding machine. In the same year, Lewis Paul, also in England, invented the weirdly named "cat carding" machine. Eventually, in 1775, Richard Arkwright, combined principles from both these machines to make his carding machine which became the industry standard. So, from about 1750, machine carding became more and more common.

Spinning remained a hand process (though assisted by a spinning wheel, rather than employing a drop spindle). The carded fiber, in the form of a sheet of 'fleece', known as a 'sliver', or a 'roving' was held in a pinch between the finger and thumb about 6 inches from the wheel, drawn out about a yard, and then allowed to relax into the revolving spindle which twisted it into a thread.

Spinning had become the bottleneck in the entire textile industry as carding became mechanized, and as the fly-shuttle loom made weaving much more efficient. By 1760, it took 6 spinners to make enough thread to feed one weaver.

The spinning jenny, invented by James Hargreaves in 1764, was still a hand operated machine. It still had a wheel. It replaced the old single, horizontal spindle with 8 upright spindles (later increased to as many as 120). Instead of drawing the roving out by hand, a wooden 'draw bar' was pulled back to draw the eight rovings out, and then slid forward to relax them into the turning spindles to make the twisted thread.

This method was inferior to hand spinning, in that control over the roving and the tension was reduced. The impact was that the thread was generally weaker and less uniform. In fact, jenny-spun thread was generally only used for weft thread, while stronger, hand spun thread remained used for warp thread. The advantage of the spinning jenny was that 8x the thread (later up to 120x the thread) could be produced for the same labor.

In 1769, Richard Arkwright improved on the spinning jenny by adding rollers, which allowed stronger thread to be produced. In about 1780, Samuel Crompton modified the spinning jenny into the 'spinning mule' through the addition of allowing the distance between the wheel and spindles to increase and decrease on rollers, giving a much better adjustment to the tension of the thread. This finally allowed mechanized spinning to truly match or exceed the quality of hand spun thread.

Source: http://saltofamerica.com/contents/displayArticle.aspx?19_271

The carding machine and the spinning jenny were followed by power looms, which made the weaving process more efficient, the cotton gin, which made short staple cotton easier and cheaper to produce in card-able form, the sewing machine, and eventually laser pattern cutters.

Before the industrial revolution, textiles were expensive and labor intensive. Building up a store of them (in a 'hope chest') was one of the main dowries of young women.

Through the industrial revolution textiles and clothing became cheaper and cheaper to the point where today basic clothing is virtually a disposable item of negligible expense in much of the world.