If $F/{\mathbf Q}$ is a quadratic field then all but finitely many places $v$ of ${\mathbf Q}$ are unramified in $F$, and we could interpret what that means in a couple of ways: prime ideal factorization (for nonarchimedean $v$), extensions of absolute values (any $v$), or base extension by ${\mathbf Q}_v$ (any $v$). Let's use the last way: we look at ${\mathbf Q}_v \otimes_{\mathbf Q} F$. First suppose $v$ is nonarchimedean. For unramified $v$, the tensor product is ${\mathbf Q}_v \times {\mathbf Q}_v$ (if $v$ splits) or it is an unramified quadratic extension of ${\mathbf Q}_v$ (if $v$ is inert). For ramified $v$, the tensor product is a ramified quadratic extension of ${\mathbf Q}_v$ in the sense of ramified extensions of complete discretely valued fields. Next suppose $v$ is the archimedean place of ${\mathbf Q}$. We call $v$ unramified in $F$ when ${\mathbf Q}_v \otimes F$ is ${\mathbf R} \times {\mathbf R}$ ($F$ is real quadratic), and call it ramified in $F$ when ${\mathbf Q}_v \otimes F$ is $\mathbf C$ ($F$ is imaginary quadratic).

When $B$ is a quaternion algebra over ${\mathbf Q}$ and $v$ is a place of ${\mathbf Q}$, the base extension ${\mathbf Q}_v \otimes_{\mathbf Q} B$ is the matrix algebra ${\rm M}_2({\mathbf Q}_v)$ except for finitely many $v$, when it is a division algebra (something subtle). Think of ${\rm M}_2({\mathbf Q}_v)$ as being a noncommutative analogue of ${\mathbf Q}_v \times {\mathbf Q}_v$, and this is a reason for calling ${\rm M}_2({\mathbf Q}_v)$ the unramified case while the division algebra case is called ramified since those are the finitely many peculiar cases. The situation with quaternion algebras doesn't have anything like the inert case from quadratic fields, so "split" and "unramified" are used as synonyms (we say $v$ is "split" or "unramified" in $B$ when ${\mathbf Q}_v \otimes_{\mathbf Q} B$ is a matrix algebra, and "nonsplit" or "ramified" otherwise).

An analogy with quadratic extensions $L$ of $K :={\mathbf C}(X)$, where all residue fields are algebraically closed, is useful to bear in mind. For a discrete valuation $v$ on $K$ that is trivial on the constants, there is a notion of ramified or unramified for $v$ in $L$. It could be defined in terms of how the prime ideal associated to $v$ in the localization ${\mathcal O}_v$ of $K$ decomposes in the integral closure of ${\mathcal O}_v$ in $L$. Or it could be defined in terms of the base extension of $L$ by the completion of $K$ at $v$, which is $K_v \otimes_{K} L$: if this is $K_v \times K_v$ then we call $v$ unramified or split in $L$ (the two words are synonyms here) and otherwise this base extension is a quadratic ramified extension of $K_v$ (in the sense of ramified extension of a complete discretely valued field) and we then call $v$ ramified or nonsplit in $L$.