Will a 650x23 tire fit on my 26" rim? I have an MTB that I have been riding in the city for many years. As such, slick tires suit me well and I have been using a set of 26x1.25 tires for the last 3-5 years but they are starting to show their age, especially after I got caught in the rain last week and the water cleaned off all the dirt and dust. I noticed a number of small punctures that looked like they were stretching out as well as what I can only describe as spider veins (though not cracked)

For some reason, 26" slick tires in widths less than 1.5 are harder and harder to find locally nowadays. I saw one tire listed online as 26x1.0 / 650x23. Are these size equivalent?
There are around 8 different rim sizes that fit tires all called "26 inch." Many are obsolete. The tires all have different Bead Seat Diameter (BSD) and are thus not interchangeable.

The ones you are most likely to see in your local bike shop are as follows (largest to smallest):

26 x 1-3/8" = BSD 590mm. Often found on middleweight 3-speeds.

650B (aka 26 x 1-1/2") = BSD 584mm. Mostly found on French or French-style touring or sport cycles. It has found new life as the new "27.5"er MTB size.

650C (aka 26 x 1-3/4") = BSD 571mm. This uses about the same rim diameter as the 26" tubular race tire. Occasionally used on small racing frames.

Last but not least, 

26" mountain or cruiser (aka 26 x 1.75") = BSD 559mm. THIS is the most common size in the USA for the last seventy years, and the size you need.

You can ignore the old inch-based designations if you wish to avoid confusion, and use the ISO sizing standard instead. It is printed on the side of every tire made since about 1964, and consists of two numbers: the tire width in mm, and the BSD in mm. So if you want a 26 tire that fits your mountain bike rim but is only an inch wide, you should look for a tire marked 25-559. 

Personally, I do not like tires that small for city riding, as they must be kept at very high pressure, and even then, they are prone to pinch flats. I prefer tires at least 1.25" (32-559), and am riding a 1.75" (47-559) tire right now.