How can i check my internet speed , and how can i check my internet is running or not.

Here are a couple of good speed tests.
Speedtest.net - The Global Broadband Speed Test
Bandwidth Place Speed Test

Check your network connections in Windows to see if you are connected to a network.

Firstly are you on dialup or broadband adsl 1, adsl 2, adsl2+, or a cable connection as these all have different speeds.

Those speed test sites are not accurate enough to really give you the true download speed.

I can download a 700 meg Linux distro in 6 minutes on my ADSL2+ connection of 15,424 Kbps...

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Wombat's: Bulldozers of the bush...

Speedtests are not that accurate and most are a bit over reaching if you ask me. If you are promised by your ISP that you have a certain download/upload speed range then ask them what test is the one to use for verification!

The ones that S&M gave you are the most popular and can give you an insight to what results that you have and how accurate they are really depends on where the test site is located and to what proximity to you it is.
It also has a lot to do with what programs that you have running on your computer....such as anti-virus and malware/spyware programs will slow down your download capabilities among what other programs are using the bandwidth as well.

And one thing the ISPs really hate to admit to is if you are using your computer during prime time...you will be much slower than advertised. The ISP's promise you high download speeds but fail to tell you that you are on shared bandwidth with your neighbors and whom else is within that particular section of the IP range. and so when you are all on at the same time...well your speed is much slower.

You can check your internet speed here in this site Your IP :: IP Tools :: test internet connection speed :: test bandwidth