403(b) Plans are most commonly used for teachers and employees of public school systems, charitable organizations and certain ministers.  In addition,  teachers also may be offerred a defined benefit pension plan. 
The maximum amount that you can defer in 2018 is $18,500.  If you are at least 50 years of age in 2018, your 403b contribution can be increased by $6,000 to $24,500.
There are many complexities associated with 403b Tax Sheltered Annuity plans. Even though a 403b plan resembles a 401k plan, there are many distinct differences.

This discussion is not meant to cover all of the nuances of a 403b plan. You should seek competent advice on 403b issues that you do not fully understand.
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403b Tax Sheltered Annuity Plans-TSA

403b Limits For 2016 - 2017
403b Plan Rollovers

You cannot set up a 403b plan yourself, it must be set up by a qualified employer.  The "403(b)" refers to the Internal Revenue Service (IRS) code section that contains the laws for Tax Sheltered Annuities.  Some attributes of a 403b plan closely resemble the characteristics of a 401k salary deferral plan. 
The easiest way for a rollover is a direct transfer from your 403b trustee to the new trustee of the other plan -- no income tax withheld. 

If you elect to directly receive the assets of your 403b account, the IRS requires that 20% be withheld for income taxes.
You then have a 60 day time period in which to complete a tax-free rollover. However, you will only possess 80% of your 403b account, which will require you to use other funds to complete the 100% rollover.

A direct transfer is much easier and avoids any potential income tax traps.
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403b Plans - Tax Sheltered Annuity (TSA)
403b Rollovers
Your 403b plan is eligible for a tax-free rollover to another retirement type plan.  An eligible retirement plan would include another 403b plan; an IRA; or another qualified retirement plan.

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How Much Can I Contribute To My 403b Plan In 2018 & 2019?
Whereas 403b plans are generally for non-profit organizations, 401k plans are for profit making businesses. Through a salary reduction agreement with your employer, you can elect to defer a part of your salary that is then contributed to your 403b account.
You can also rollover a qualified retirement plan into your 403b plan.   For instance, assume you were in a 401k plan when you had a job as a salesperson.

If you later decided to become a school teacher, you could rollover your old 401k into your new 403b plan offered by your school district.
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These are the same limits as for a 401k plan.

The 403b maximum  amounts for 2019 are increased to $19,000 and if age 50 the limit is $25,000

There additionally is a 15-Year Rule that can increase your yearly 403b contribution. 
To qualify for this additional contribution, you must have at least 15 years of service with your qualifying employer.  Your extra 403b contribution under the 15 Year Rule can be as much as $3,000.