There's 2 questions in your question!
Hello is have prepared a business card using cmyk mode keeping
resolution as 300. Now while exporting the image as jpg the colors get
brighten up.....how to solve?
I have kept proof setup on and in color picker the 'only web colors'
check box is off how to recognise which proper tint shade of color
will not get brighten up after exporting?
To keep your CMYK colors, use the "save as" to save your jpg, not the "save for web."
The "save for web" optimizes your image, remove the resolution data and convert it to use only the pixels at low resolution. It also changes the color mode to RGB. That's not something you want for a print project.
The "save as..." will save your jpg using the color mode your were working in: RGB or CMYK, and will also keep the resolution (300ppi), actual size and you can also embed your color profile with it by checking the box on the "save as" window. This can work for your project.
It's also normal the colors will look different if you open the jpg in a web browser for example or if your color profile was or wasn't added to the jpg.
If you're not certain if your jpg is alright, simply open it back in Photoshop, verify the color mode (usually, it's indicated on the top of the jpg window next to the bits) and verify your resolution. If everything is alright, your jpg is really a CMYK @300ppi.
Some printers are ok with jpg although it's not the best print quality you can get, but for digital printing it's ok (eg. small quantities in full colors like VistaPrint.) But if you have the option of sending a PDF with your text in vectors, it's even better.
See this link to see how to convert your PSD to keep your text layers in Vector:
How to keep the text in vector in Photoshop without rasterizing it or flattening the layers when exporting to PDF?
Usually while selecting the color from color picker we have two boxex
new and current. Sometimes Besides new a small triangle icon appears what does that mean?
The little triangle you see means the color you see is not "print friendly" and the color swatch below it is a suggested color that is within the "print gamut" (CMYK "print friendly".) If you click on the suggested swatch below it, it will simply show you a realistic preview of that CMYK color.
If you see that warning, make sure to verify that your color mode IS CMYK and not RGB. Go in the menu "Image" then "Mode" then select "CMYK." If you are already in CMYK color mode, the suggested color is usually not very different (or not different at all) from the current color. You can see it by looking at the CMYK color values on the right.