Health insurance lines on a payslip - explanations

I have Barmer as my insurance provider. This was setup for me by my employer the first week I was here. I really was not offered any other choices. What I wanted to know is , are their cheaper alternatives? Is there a goverment insurance, I assume Barmer is private.

Also can someone explain these lines on my payslip

AG-Ant. freiw.Krankenvers +243,23
Gesamtbetr. freiw.Kranken -518.19

Maybe these 2 are not health insurance related
AG-Ant. Pflegevers.b.frei +29.96
Gesamtbetr.Pflegev.b.frei +59.92

Do these amounts seem reasonable for a family with kids?

Barmer is one of the standard health insurers. often you'll just get put in a health insurance with whoever your employer usually uses. In your case it's Barmer.

The 2 things you mention are health cover and care cover.
You are paying half and your employer the other half.

Seeing as the sum covers you and your dependents it's a pretty good deal, last time I looked I pay that for just me! Families really do well out of the public system whereas singles can save going totally private.

Barmer is one of the "public" insurers. The amounts you pay for your coverage are calculated as a percentage of your income. These percentages differ by insurer, but the differences are usually not large. You can look around if you find a cheaper insurer (you can choose any public insurer you like, and they cannot refuse you. They charge the same percentage fee for everyone), but even if Barmer happens to be one of the more expense insurers (I do not know if it is expensive or not), you would at best save 5-10 %.