What I'm trying to understand, what is special about Ethernet specifically that he was chosen to be
the primary protocol for our personal computers by default.
Ethernet (IEEE 802.3) has been dethroned, and Wi-Fi (IEEE 802.11) is now King of the LAN. There are more devices shipping with Wi-Fi than there are shipping with ethernet.
Still, IEEE 802 has other protocols to be configured for our Model OSI
Layer 2. So the reason for the cost still doesn't really "bought me".
Having lived through the wired network wars, I have some personal perspective on this. I remember when ethernet interfaces were expensive. ($750 each, of course, the computer you wanted was $5000, and that was a lot of money 30 years ago). There were many different LAN interfaces/protocols used: G-NET, ARCNET (IEEE 802.4), token ring (IEEE 802.5), etc. Some were cheaper and some more expensive than ethernet, but basic ethernet was faster at 10 Mbps than the others, like token ring (4 Mbps) and ARCNET (4 Mbps), which were also expensive and saddled with patent royalties. Token ring and ARCNET had the advantage of no collisions, but the landscape changed with the introduction of ethernet switches. Also, ethernet was starting to be produced by companies other than 3Com (creating competition and lower prices), whereas token ring and ARCNET were still basically tied to single companies and expensive intellectual property royalties. Ethernet was pushed to 100 Mbps when token ring got to 16 Mbps, and Thomas-Conrad finally bumped ARCNET to 100 Mbps, but it was late to the party. Token ring and ARCNET were finally ported to UTP, which was cheaper and easier to use than the coax used by the original ethernet and ARCNET, or the proprietary cabling for token ring, but they were simply following ethernet. In the end, it really boiled down to lower prices, and the fact that ethernet could be mass produced by many different companies on single chips.
the specification for hardware(RJ-45 Cable)
You are confused. There is no such thing. RJ-45 is a type of keyed connector for voice. UTP LAN cabling uses 8P8C connectors that are often called RJ-45, but that is incorrect. In any case, that is not the cabling. Ethernet runs on a variety of media. The common medium for LAN is UTP, which is classified by ANSI/TIA/EIA as categories (Category-3, Category-5e, Category-6, and Category-6a are the only currently registered categories), or by the ISO/IEC classes, which includes shielded copper cabling, too. UTP was cheap, plentiful, and easy to install. Ethernet, at 10 Mbps, ran on Category-3 cabling used by voice, so it allowed for a single cable plant for a business. Faster LAN speeds required different UTP cable categories (Category-4, which no longer exists, for 16 Mbps token ring, and Category-5 which no longer exists, replaced by Category-5e, for 100 Mbps ethernet and ARCNET, Category-5e and Category-6 for 1 Gbps ethernet, and Category-6a for 10 Gbps ethernet). All the newer cable categories are backwards compatible, so you can still install a single cable plant for voice and data, and any UTP cabling to a WA (Work Area) can be used for either voice or data.
About carrier ethernet, i read on wikipedia and other online forums
that it provides ethernet capabilities for the carrier world(STM, ATM,
SDH, Sonet, etc...) I didn't see as well any intel about why carrier
ethernet was chosen , or to be precise was created.. The only reasons
I have found was the cost reason but i didn't found technical answers
The telcos were/are using the TDM technologies that were designed for voice, and we were saddled with expensive hardware (CSU/DSU) to put data on the telco lines. You also need different equipment, depending on the speed (T1, T3, etc.). Data exploded on the scene, especially after the Internet went commercial in 1995. Companies need to move increasingly large amounts of data over long distances.
The introduction of VoIP really got the telcos attention, and they are all moving toward that. A single protocol for voice and data is very attractive, and POTS calls are now often using VoIP in the carrier networks. The problem is the massive installed base of TDM, and the time it takes to convert it to ethernet. In many cases, the ISP delivers ethernet to the customer, it is really on TDM circuits with a conversion box that gives ethernet to the customer, while taking advantage of the existing, installed TDM circuits of the telcos. That is changing, but it takes time. The advantage to the customer is that the CE equipment can use a single protocol, albeit a mixture of media (copper and fiber). Fiber has the advantage of much longer distances at higher speeds than the UTP cabling (limited to 100 meters) installed for the WAs. The price for fiber and copper cabling and installation are now competitive, but the interface costs are still much more expensive for fiber, so fiber is used for the LAN backbone and WAN, where distance is a problem for copper.