Buy Japan Pottery or Ceramics and Japanese Cloth

How about Aichi-Prefecture !

http://japan-guide.com/e/e3004.html
I would recommend Asakusa area where there is a famous temple. The above link would give u a brief introduction of the area.
Along the road from the train station to the temple, there r many small old shops & restaurants. Most of them have been there for decades & even over a century or longer.
There is a 2-storey ceramics shop at a corner that u can find a lot of wonders. Handmade tabelware from very affordable US$3 up to more interesting designs around US$35 up to artpiece from famous ceramist that can cost over US$10,000. Even not buying, they r adorable to look at. I just bought a lot back home from there earlier this year. On the way walking on the main street from Asakusa to Akihabara, there is a big 2-storey one that sells more younger designs from various companies of different perfectories. The company would have their signature/ stamp at the bottom of the tableware. Each company would have their own design / glazes characteristics. So it is fun to have a chance to browse through collections coming from different areas.

For Japan clothing, do not buy inside the Asakusa temple. Hop into the small streets around the temple on the sides, there r a couple of old shops selling kimono & traditional clothes there if u mean Kimono. But it is usually very very expensive.
If u mean Japan fashion brands who sell made in Japan clothes, suggest u can try big deparment stores like Isetan & Daimaru. Look at the label to check if the clothes r made in Japan (I always do this & u would find that those ''made in Japan'' r really better in terms of cutting, fabric, etc.). Now most Japan brands sell clothes made in anywhere in the world, like India, Turkey, etc.
If u like fashion, Aoyama & Daikanyama sell good designers' clothes but a bit expensive if not on sale.

My favourite items to buy in Japan is
- handkerchieves
- scarves
- leather gloves
- umbrella
- wallet
- sockings

Make sure that they r made in Japan. Usually they offer good value & quality (even better if it is on sale. Sale seaon is Jan - early Feb & July - early Aug). There r many exclusive designs for Japan only even run under international brandnames like Givenchy, Burberrys, Vivienne Westwood, etc.

Enjoy ur stay & shopping there.

You can purchase Japanese pottery in just about ANY city (and yes, it is authentic). If u visit Tokyo or Kyoto there will of course be many stores from which to choose. All department stores carry pottery from different studios, the cheap stuff is made in China, the more expensive stuff is made in Japan. There is also a store called Oribe (it is a chain) that is in most malls; they have a wide selection of pottery & other crafts. In my town the supermarket would often have vendors come & sell things in the parking lot or vestibule (not unlike a mini flea market). Some of the sellers were selling pottery they'd made themselves, & it was very cheap & beautiful!
Another great place to find pottery is at the recycle shop (thrift store). Japanese people always receive pottery at weddings & many people either do not have space for it, do not appreciate it, never use it, etc etc. So u can find things NIB for about 1/10th the department store price. (For example, I once got a brand new Le Creuset pot there for about $10! In a dept store the same pot was over $100. Le Creuset is not pottery, but u get the point.)
Fabric is not as easy to find. I lived in a smallish city & most of the fabric you'd see for sale was precut into 1m lengths. Dept stores & craft stores sell this type fabric for about $10/m. It is not the super high-end stuff, but it is of high quality & the some of the designs r so cute--everything from kids' characters to traditional Japanese designs, etc. There r also longer lengths sold in summer for making yukata. I accumulated quite a lot of it. . .! In my town there was ONE large fabric store that was heaven on earth. Japan makes some of the MOST beautiful fabric u have ever seen, & it is not cheap. That store is called Otsukaya (there is more than one).
The previous poster listed the most famous studios in Japan--if u r a collector, u can research each one & see which is most to ur taste (''-yaki'' btw could be translated here as ''-ware''). These items r quite expensive; the cheapest places to find them r in their town of origin. But if u r just a tourist looking to find something nice, u can take my advice too.

u better go one of that pref. & visit kamamoto (pottery)
you can find very good & cheap one.
lots of design.

mashiko-yaki (pottery) (dark red, green, black)
mashiko, tochigi pref.

arita-yaki = imari-yaki (porcelain/china) (veeery famous! white ground, blue picture)
arita, saga pref.

kiyomizu-yaki (porcelain/china)
kyoto pref.

kutani-yaki (porcelain/china)
kaga, ishikawa pref.

shigaraki-yaki (pottery)(very famous! dark brown)
siga pref.

hagi-yaki (pottery)(i love this!)
yamaguchi pref.

bizen-yaki (pottery)
okayama pref.

*add
about japanese made cloth
aizome(indigo) is very famous (tokushima pref. but u can buy anywhere in japan)
http://shop.ebay.com/items/japanese-aizome_W0QQ_nkwZjapaneseQ20aizome