Yes, this is a rollover or a direct transfer.
You can move a 403b to an IRA. If it is pre-tax funds, it will move to a traditional IRA, if it is a Roth-option after-tax 403b, it will move to a Roth IRA. You want to do a direct transfer, from one trustee to the other--open the IRA, then give the IRA account information to the 403b trustee to do a direct transfer. The direct transfer is not subject to tax or penalties and you can do as many direct transfers as you like.
You could do an indirect rollover, meaning that the 403b sends you the funds by check or electronically, and then you send the funds to the IRA. But you have a 60 day time limit, or else the 403b withdrawal is taxable and subject to penalties.
The rules for a 401a are basically the same; pre-tax money can be rolled over to a traditional IRA and after-tax money can be rolled over to a Roth IRA. Again, you want a direct transfer if you can arrange it.
You can rollover pre-tax funds into a Roth IRA, but that also counts as a conversion and will be subject to income tax, but not a penalty.
There is a chart here. https://www.irs.gov/pub/irs-tege/rollover_chart.pdf
