Dobsonian telescopes are really Newtonian design optics on a special alt-az mount popularized by John Dobson. Alt-az stands for altitude and azimuth. Altitude is up-down and azimuth is side-to-side. That describes how you move the scope.
A top end Dobsonian will cost around $300 per inch of aperture. A top end refractor will cost around $1000+ per inch of aperture. So, a five inch refractor and a fifteen inch Dobsonian cost the same. The Dobsonian by definition *includes the mount.* People who buy high end refractors have to *pay additional high costs for the right mount.*
So the Dobson is an extremely cost effective design. The Newtonian design is very effective at controlling cost, and as a matter of optics, is perfectly controlled for color aberration. The Dobson mount typically is highly leveraged, since the long tube rotates on "trunions" that are at the base of the scope. Most of the weight of the Dobson Newtonian is in the mirror.
This extremely cost effective design revolutionized amateur astronomy and made it possible for individuals to own mirrors as large as thirty or thirty five inches. Forty years ago six inches was the standard Newtonian telscope, and the person who came to a star party with a ten inch was "God."
All this said, the defect of the design is that as aperture increases the eyepiece rises from the ground, and you need a ladder. The mechanical considerations force a short focal ratio which leads to an optical aberration called COMA, a blurring of stars off axis. Dobsonian telescopes range from Chinese imports for a few hundred bucks to high end things of beauty for several thousand.
It is the same with refractors: cheap ones can be had for several hundred dollars. But as a matter of engineering you can't make a consumer refractor that is as large as a consumer Dobson. So the Dobson telescope has the advantage of APERTURE. It will bring in dimmer objects at lower cost. But as size increases various tradeoffs set in, you may need a ladder or stool to reach the eyepiece, and the mirror often takes so long to cool down at night that the heat waves compromise performance. CHEAP refractors do not use the exotic glasses and three-lensed designs that reduce color. EXPENSIVE refractors require precision lens grinding by an optical master and the use of extremely expensive, exotic glass.
Refractors have the major advantage of WIDE FIELDS, FASTER COOLDOWN, and precision optics. The high end ones are well suited for astrophotography. They almost always are used on German equatorial mounts and have tracking. The cheap refractors will show false color on bright objects (chromatic aberration) that cheap Newtonians will not have. Refractors typically require less work to keep usable: they seldom need to be collimated, which is a routine ritual for the newtonian owner.
Before there were Dobson Newtonians, Newtonians were typically sold on heavy cast iron piers with German equatorial mounts (see link). The mount design limited the size of the mirror and the cost of the mount further restricted how much money could be put into the optic. The simplicity and cheapness of the Dobson wooden box mounts provided a simple way to give more stability to the Newtonian mount.
Today, among high end telescope users, one sees Newtonian telescopes on equatorial mounts only infrequently. (A lot of the cheap scopes are sold on very wobbly equatorial mounts that probably should be avoided.) The Schmidt-Cassegrains and refractor designs tend to go onto equatorial mounts.
Although the Dobson mounts can and often do have tracking, the design is not favored for astrophotography because of coma, because the leveraged eyepiece end of the scope gets wobbly with heavy equipment on it, and because tracking on an alt-az telescope fails to correct a problem called FIELD ROTATION which will cause stars to streak in long exposures.
This is as much as I can do in this short format. All of these telescopes should be thought of as "engineering solutions" which integrate such factors as optical aberrations, mount stability, cost, size of aperture, size of field of view, into one instrument. As a result the question is somewhat like asking whether a RAM Dodge pickup is better than a Lamborghini. The pickup is good for one set of applications, the Lambo for others.
Amateur astronomers love to gas on about which design is best, but the fact is that there is no unique solution to the problem of gathering light and making it available to an observer or camera. There are only plusses and minuses, costs and benfits.
If you want to know what to get for yourself, the answer is, get a 6 to 8 inch dobson from a place like Orion Telescopes for three or four hundred bucks (for example, the Orion Skyquest XT8). Or buy them at about half price on astromart (used: you will make back your $12 one-time lifetime joining fee on your first purchase). If you are in the middle age crisis zone and want to put some bucks down on really beautiful equipment that, altogether, will cost considerably less than a Porsche, then you can investigate the Takahashi TOA 130 or TEC 140 (among refractors) on an Astro-Physics GTO900 mount and the Obsession Telescopes 15" (or a 25" if you are feeling ambitious). But the cheap Chinese imported Newtonian on a Dobson style mount will show you a lot of sky, and do it a lot better (with more aperture and no color aberration) than a cheap chinese imported refractor. For any given cost, the larger aperture (diameter) is usually the way to go (except for people who are totally in love with refractors) because the scope will gather more light and show you more details on planets and more stars and galaxies in deep space.
To understand why people fall in love with refractors, you need to own one.
Hope that helps,